I can state what I believe are the Eastern Orthodox (my faith) answers to your questions.
For example, one commonly stated purpose of the church is to
evangelize the lost. But is that actually necessary in a wider
hope/inclusive framework? (see related question) If not, then what is
The Church (capital "C") is what Christ established for us on earth in order for us to work out our salvation (Philippians 2:12), making use of the Grace-given means of sanctification provided by it. It is where all are gathered together in Christ (Ephesians 1:10).
The Orthodox Church has a much different understanding of sin, salvation, the nature of man, and his relationship to God, than the Roman Catholic and most, if not all, Protestant Christian confessions. We believe in ancestral sin, which has damaged our nature, but we do not believe in ancestral guilt. Thus, salvation is seen more as a process of healing - the word "save" in Greek really means "heal" - than of escape from punishment. It is, if you will, a faith of restoration rather than retribution.
Orthodox believe, furthermore, that although we are only saved by virtue of God's grace, we must cooperate with that grace - a process known as synergy. This cooperation of man's free will with God's grace was a subject of somewhat bitter debate in the early 4th century between Augustine and many (but not all) western bishops in the west, who held that man cannot do anything at all to enable his salvation; and other Church Fathers such as John Cassian who took the view that man cooperates with God. The Augustinian view logically led to a somewhat dismal theology of predestination and salvation that has persisted to this day in the west.
To your statement,
In the wider hope/inclusivist theological framework, very simply put,
people who have not heard the Gospel are not necessarily lost and
destined for hell.
I would say that this framework does not have a solid foundation in Scripture, unless one were to construct a logical chain of very specific pericopes. The Gospels are very straightforward and very clear. Unfortunately - in my opinion - western Protestant theology has tended to rely on interpreting the Gospels through the writings of Paul, rather than interpreting what Paul wrote through the Lord's own words. The Lord Himself stated very clearly who would be lost and who would be saved:
When the Son of man comes in his glory, and all the angels with him, then he will sit on his glorious throne. Before him will be gathered
all the nations, and he will separate them one from another as a
shepherd separates the sheep from the goats, and he will place the
sheep at his right hand, but the goats at the left. Then the King will
say to those at his right hand, ‘Come, O blessed of my Father, inherit
the kingdom prepared for you from the foundation of the world; for I
was hungry and you gave me food, I was thirsty and you gave me drink,
I was a stranger and you welcomed me, I was naked and you clothed me, I
was sick and you visited me, I was in prison and you came to me.’ Then
the righteous will answer him, ‘Lord, when did we see thee hungry and
feed thee, or thirsty and give thee drink? And when did we see thee a
stranger and welcome thee, or naked and clothe thee? And when did we
see thee sick or in prison and visit thee?’ And the King will answer
them, ‘Truly, I say to you, as you did it to one of the least of these
my brethren, you did it to me.’ Then he will say to those at his left
hand, ‘Depart from me, you cursed, into the eternal fire prepared for
the devil and his angels;for I was hungry and you gave me no food, I
was thirsty and you gave me no drink,I was a stranger and you did not
welcome me, naked and you did not clothe me, sick and in prison and
you did not visit me.’ Then they also will answer, ‘Lord, when did we
see thee hungry or thirsty or a stranger or naked or sick or in
prison, and did not minister to thee?’ Then he will answer them,
‘Truly, I say to you, as you did it not to one of the least of these,
you did it not to me.’ And they will go away into eternal punishment,
but the righteous into eternal life (Matthew 25:31-46)
In short, the Lord is stating how important love of our neighbor is for our salvation - something we see him emphasizing elsewhere in the Gospels (e.g. Matthew 22:34-40; Mark 12:28-31).
Why should I "go to church", one might ask, if according to the parable it doesn't even seem to matter whether I believe in Christ, or even God? I would answer this by saying that given how the devil, the world, and our own sinful nature conspires against us, we cannot succeed in loving our brothers and sisters without the grace of God. Some of us have a great gift available to us - the Church - that helps us to heal our souls and learn from the Lord Himself (viz. Matthew 11:29). Others of us, however, do not for whatever reason have this gift available. We must recall, perhaps with fear and trembling, however, that the Lord told us that we who have His Church available to us will be held all the more accountable for how we lead our lives. Every one to whom much is given, He told us, of him will much be required (Luke 12:48).