Hooker wrote in the last years of Queen Elizabeth I. He feared that the Church of England, as he knew it, was under threat. The Puritans were growing in strength and influence. He was amongst the many, referred to in the Epistle Dedicatory to the King James Bible, who feared that when Elizabeth died, thick and palpable clouds of darkness would cover the land. He wrote to defend the Church of England as it then was. As things turned out the Puritans did not achieve power until the middle years of the next (i.e. 17th) century. Following the Restoration of King Charles II in 1660 the Church of England returned, in the main, to the way it was in the reign of Elizabeth and Hooker was a major influence on the Caroline Divines and future generations of Anglicans around the world.
Though for no other cause, yet for this, posterity may know we have not loosely through silence permitted things to pass away as in a dream; there shall be for men's information extant thus much concerning the Church of God established among us, and their careful endeavours who would have upheld the same.
As the OP points out Hooker believed every Church, that is to say every Christian community (e.g. England), had the right to make its own rules. This obviously excludes the idea that rules be imposed from Rome. But that arguent had already been won in England, it was not Hooker's focus. Hooker was principally arguing against the Regulatory Principle, the idea that Scripture contains all that is needed concerning worship and church order.
In Book Three he says:
We have asserted that there is a difference between a) matters of faith which are necessary to salvation; and b) ceremonies and the form of Church government. The first we say must be collected from Scripture. For the second, it is sufficient that there be nothing in Scripture against them.
Referring to Germany (Luther) and France (Calvin) he says:
When Germany had stricken off that which appeared corrupt in the doctrine of the Church of Rome, but seemed still to retain a conformity in discipline; France took away the Popish orders which the Germans retained.
Whee Luther trimmed what was wrong, Calvin tried to start again, as if from scratch.
In this England was similar to Germany, and Scotland was similar to France (and Geneva). The Church of England was accused by the Puritans of retaning far too many practices of the Roman Catholic Church. It had not yet been as fully reformed as had the Church of Scotland. Hooker characterises the typical Puritan objection to pretty much anything, from wedding rings to bishops, as "We find no such thing commanded in the Word of God."
Hooker said that what was commanded in the Word of God was that nothing be done scandalous or offensive, that all things be done decently and in order, that all things be done unto edification and all to the Glory of God. Within these broad precepts the Church of England, like the Church of every other country (he had no thought of private denominations), had freedom to set its own rites, ceremonies and system of governance.
In Books Four and Five Hooker argues the case for numerous practices in the Church of England, and in the liturgy of the Book of Common Prayer. His arguments are that nothing is contrary to Scripture, and that what there is is good.
Book Six is devoted to considering whether there must be, in every congregation, lay-elders as ministers of ecclesiastical jurisdiction. He concludes there is no need for lay-elders as ministers of ecclesiastical jurisdiction, and argues very strongly against them. This might, perhaps, suggest he was opposed to lay involvement in church governance, but as we see in Book Eight, this was not the case.
What Hooker was concerned with in Book Six was repentance, forgiveness, penance and absolution. The Church of England services began with a general confession, by the congregation, and a general absolution said by the priest. There was no need for a person to confess his/her sins individually to a priest as in the Roman Church (though it was permissible). Hooker was attacking the Scottish system which, in some cases at least, required public confesion and the performance of a penance set by the Kirk Session. Every parish has a Kirk Session, consisting of lay-elders as well as ministers. It is this which Hooker means by "jurisdiction of lay-elders". He argues that it is not necessary, and that it is disadvantageous generally speaking.
Book Seven refutes the proposition that there ought not to be bishops in the Chuch indued with such authority and honour as ours are. Many Puritans believed that all ministers were equal. Hooker did not.
Book Eight refutes the arguments of those who objected to Royal Supremancy in the Church. He acknowledges that, under infidel rule, Church and State are two separate entities. But, he points out, in England Church and State are one and the same community.
As every man who is a member of the Church of England is also a member of the State, and vice versa, and the same multitude constitute both, there is no real or personal divison between them.
Hooher argued there are 2 kinds of opponents to the spiritual supremacy of the monarch. The first are Papists, those who believe in the universal jurisdiction of Rome. The second are those who believe that the spiritual power belongs to the clergy of every national Church.
Apart from the fact that Old Testament Kings exercised power over religious and civil affairs alike (Solomon the king, not Zadok the priest, built the Temple); Hooker pointed out the practical difficulties if there was no supreme authority.
Suppose that tomorrow the judicial power requires you to attend court, the martial power requires you in the field, and the spiritual power requires you in the Temple, and all have equal authority, what are you to do?
He reveals what he thinks of lay governance in the Church in the following passage:
The Parliament of England, with the Convocation annexed, is the essence of all the government within the kingdom: it is the body of the whole realm and consists of the king and (represents) all his subjects. The parliament is a court, not so merely temporal as that it should meddle with nothing but leather and wool. The most natural and religious course in makng laws is that the matter of them be taken from the judgement of the wisest in those things which they are to concern. In matters of God, in framing forms of prayer, articles of faith and religious ceremonies, the pastors and bishops of our souls must naturally be thought more fit for this offce than men of secular calling. Still, when all that the wisdom of all can devise is effected it is the general consent of all which must give the laws their form and vigour ..... And the Parliament of England has authority to give or withhold that assent and approbation which give laws their force.
The Convocation to which Hooker refers is actually two Convocations, one for the South (Canterbury povince) and one for the North (York province), each consisting of a House of Bishops and a House of Clergy.
Church Polity in the time of Hooker
Preaching the Word and administering the Sacraments was the responsibilty of priests, ordained and supervised by bishops.
The appointment of a priest to a particular parish was usually the responsibility of a layman, the patron, typically often the lord or gentleman on whose land the church stood.
Church administration at parish level, was the responsibility of churchwardens. Usually each parish had two churchwardens elected by the parishioners, and the churchwardens were always laymen.
Major developments such as the Approval of the Thirty-Nine Articles or amendments in the liturgy, or the creation of new dioceses etc was generally originated in the Convocations representing clergy in general, and bishops in particular. On approval by both houses of both convocations it would next require approval by both Houses of Parliament representing Commons and Lords (where bishops also sat); and then Royal Assent.
Churchwardens and Patrons were lay, the House of Commons members were lay and the House of Lords was partly lay. The monarch was lay. So there was quite significant lay involvement which Hooker approved of.
How Church Polity has changed
Convocations endorsed the Prayer Book of 1662 before passing it to Parliament. During the eigteenth centuriy convocations exercised almost no functions. As OP says, since 1919 there was a Church Assembly reconstitued as the General Synod in 1969. This differs from Convocation in that it also includes lay representatives These lay people must be communicant members of the Church and are indirectly elected by baptised members of the Church.
The General Synod can pass Measures, approved by the three houses (laity, clergy and bishops). If passed they are conidered by the Ecclesiastical Committee of Parliament (about 30, 15 lords and 15 members of the Commons). They will then be approved or rejected by the House of Commons and the House of Lords, and finally by the Supreme Governor of the Church of England, Her Majesty the Queen.
The House of Commons, and the House of Lords, can and do include persons of any and no religion, as well as representatives of Scotland, Wales and Northern Ireland. It is no longer the case that members of the Church of England and subjects of the UK state are one and the same group of people. The monarch must be a communicant member of the Church of England.
As OP points out there are now diocesan synods consisting of the bishop, any suffragan bishops, representatives of clergy, and representatives of laity.
At the parish level churchwardens still exist and are still elected by the parishioners. Any resident of the parish can vote, provided he or she is a Commonwealth citizen. He does not have to be a Christian of any kind to vote. The churchwardens though must be members of the Church. Alongside this there are now also parochial church councils, elected by baptised members of the Church of England living in the parish (regardless of whether they attend church) and any person living outside the parish who worships within it.
What might Hooker think?
Hooker wrote at a time when Church and Nation were identical. The Church of England today is both a national Church, and at the same time is one denomination amongst many. Many citizens are of other faiths, or other denominations, or have no interest in Church. It is a situation Hooker did not envisage or address.
Hookers principle point was that each Church has the right to make its own rules. I do not see that he would find anything impermissible in current arrangements. He would say modern England can make its rules too. But what would he think of them?
The creation of Diocesan Synods pretty much mirrored the creation of County Councils, a general acceptance in the country of the need for local repesentaion. Diocesan Synods are to bishops as County Councils were to magistrates.
The chief problem was how, and whether, to adjust for the fact that many members of civil society are not Anglicans, not Christians even.
For a time non-Anglicans were barred from Parliament. Later this became widely unaceptable and Parliament was opened up, progressively, to other Protestants, other Christians, anybody.
One solution would have been to effectively disestablish the Church, replacing the lay involvement of Parliament with a separate body consisting of lay Church members only. Another would have been to say lay Church member needd no special representation. By creating the House of Laity in the General Synod, England has recognised that the lay members of the Church are now a distinct subset of the population as a whole; and deserve a particular say. But by retaining the role of Parliament there is also the recognition that the Church belongs to, and is for, all the people of England.
At the parish level too, Parochial Church Councils, representing Church members, were brought in alongside Chuchwardens, representing everybody.
I think Hooker could have written a very strong defence of the current polity. I fear he might be less impressed with the state of the Church generally. He would, I feel sure, be pleasntly surprised that the Church of England has lasted in anything like the form he knew it; perhaps in no small part, thanks to him.