The page linked to in the question shows two versions of the Prayer over the Water which were introduced in 2015. These are alternatives, in "accessible language", to the main Prayer over the Water in the Common Worship baptism liturgy. The Common Worship baptism liturgy is itself an alternative to the Book of Common Prayer baptism liturgy which, alone, is doctrinally definitive for the Church of England.
It may be helpful to consider the historical development of the liturgy.
Prior to the Reformation baptismal water received a priestly blessing though not at every baptism. This practice was continued in Cranmer's first English Prayer Book, under Edward VI, in 1549, but removed from the 1552 edition. This is the 1549 prayer which was directed to be used whenever the water was changed before any baptism, and that the water be changed at least monthly:
O most merciful God our Saviour Jesus Christ, who hast ordained the element of water for the regeneration of thy faithful people, upon whom being baptised in the river of Jordan, the Holy Ghost came down in the likeness of a dove: Send down we beseech thee the same thy Holy Spirit to assist us, and to be present at this our invocation of thy holy name: Sanctify (+) this fountain of baptism, thou that art the sanctifier of al things, that by the power of thy word, all those that shall he baptized therein, may be spiritually regenerated, and made the children of everlasting adoption. Amen.
The (+) indicates the sign of the Cross to be made. I have modernised the spelling.
This practice was removed by Cranmer from the 1552 and subsequent versions, probably influenced by Martin Bucer, who died in exile in England in 1551, and who strongly opposed blessing inanimate objects.
Also in 1549 was introduced a version of Luther's Flood Prayer, as the first prayer in the English baptism liturgy. This was modified in 1552, in the Second Prayer Book of Edward VI, and again (as we will come to) in 1662. This is the 1552 version:
Almighty and everlasting God, which of thy great mercy didst save Noah and his family in the Ark from perishing by water: and also didst safely lead the children of Israel, thy people through the Red Sea: figuring thereby thy holy Baptism; and by the Baptism of thy well-beloved Son Jesus Christ, didst sanctify the flood Jordan, and all other waters, to the mystical washing away of sin: We beseech thee for thy infinite mercies, that thou wilt mercifully look upon this child, sanctify him and wash him with thy Holy Ghost, that he, being delivered from thy wrath, may be received into the Ark of Christ's Church, and being steadfast in faith, joyful through hope, and rooted in charity, may so pass the waves of this troublesome world, that finally he may come to the land of everlasting life, there to reign with thee, world without end, through Jesus Christ our Lord. Amen.
The relevant bit here, which I have bolded, is that, Christ's baptism sanctified not only the Jordan but all other waters for baptismal use. This is also in Luther's original Flood Prayer. It was un-necessary for the priest to bless the water, because all waters were sanctified already.
Edward, known as England's Josiah, died aged 15 the following year,1553, and his older sister Mary undid his reforms and England reverted to the pre-Reformation use. On Mary's death in 1558 Elizabeth restored most of the 1552 book, including the bits relevant to this question.
In 1645, during the Civil War Era, Parliament approved, without the King's authority, the Directory of Worship, which was to replace the Book of Common Prayer. This re-instated the blessing of baptismal water. In keeping with the general concept of the Directory there was no prescribed form of words. This is an extract from the section on baptism:
Prayer is also to be joined with the Words of Institution for sanctifying the water to this spiritual use.
Following the Restoration it was decided to re-introduce the Book of Common Prayer but with suitable amendments. A Conference, the Savoy Conference, was held to discuss these. It comprised Presbyterian Divines and Church of England Bishops. The format was that the Puritans made a list of the exceptions (objections) they had to the Prayer Book, and the Bishops responded.
One Exception was to the phrase in Cranmer's version of Luther's Flood Prayer:
By the Baptism of thy well beloved Son, etc. didst sanctify the flood Jordan, and all other waters to the mystical washing away of sin.
The exception was:
It being doubtful whether either the flood Jordan, or any other waters were sanctified to a Sacramental use by Christs being baptized and not necessary to be altered: we desire this may be otherwise expressed.
The Bishops' reply was:
If Jordan and all other waters be not so far sanctified by Christ, as to be the matter of Baptism, what authority have we to baptise?
To which the Puritans replied:
Did Christ sanctify all Corn or Bread, or Grapes or Wine to an holy use, when he administered the Lords Supper? Sanctifying is separating to an holy use; But the flood Jordan and all other water, is not separated to this holy use, in any proper sense.
The consequence of this discussion was firstly a change in the Flood Prayer, and secondly the introduction of a blessing of the water in the last prayer before the baptism itself.
In the Flood Prayer:
..by the Baptism of thy well-beloved Son Jesus Christ, didst sanctify the flood Jordan, and all other waters, to the mystical washing away of sin
became
by the Baptism of thy well-beloved Son Jesus Christ, in the river Jordan, didst sanctify Water to the mystical washing away of sin.
The last prayer before the baptism itself was amended by adding the phrase in bold:
Almighty ever living God, whose most dearly beloved Son Jesus Christ, for the forgiveness of our sins, did shed out of his most precious side both water and blood; and gave commandment to his disciples, that they should go teach all nations, and baptize them In the Name of the Father, and of the Son, and of the Holy Ghost: Regard, we beseech thee, the supplications of thy congregation; sanctify this Water to the mystical washing away of sin; and grant that this Child, now to be baptized therein, may receive the fullness of thy grace, and ever remain in the number of thy faithful and elect children; through Jesus Christ our Lord. Amen.
So the doctrinally definitive version of the baptism liturgy was deliberately altered to make ambiguous the phrase in the Flood Prayer as to whether all water, or water in general was sanctified for Baptism, and to very specifically introduce in another prayer a ministerial blessing of the particular water intended for use in a particular baptism. Perhaps surprisingly, this was done at the request of the Puritans, rather than the High Church side.
The Alternative Service Book, available as an alternative in the 1980s and 1990s, combined the flood and the last prayer into a Prayer over the Water as follows:
Almighty God, whose Son Jesus Christ was baptized in the river Jordan
we thank you for the gift of water to cleanse us and revive us;
we thank you that through the waters of the Red Sea,
you led your people out of slavery to freedom in the promised land;
we thank you that through the deep waters of death you brought your Son,
and raised him to life in triumph.
Bless this water, that your servants who are washed in it
may be made one with Christ in his death and in his resurrection,
to be cleansed and delivered from all sin.
Send your Holy Spirit upon them to bring them to new birth
in the family of your Church,
and raise them with Christ to full and eternal life.
or all might, majesty, authority, and power are yours, now and for ever.
And the main Prayer Over the Water in the Common Worship goes:
We thank you, almighty God, for the gift of water to sustain, refresh and cleanse all life. Over water the Holy Spirit moved in the beginning of creation. Through water you led the children of Israel from slavery in Egypt to freedom in the Promised Land. In water your Son Jesus received the baptism of John and was anointed by the Holy Spirit as the Messiah, the Christ, to lead us from the death of sin to newness of life. We thank you, Father, for the water of baptism. In it we are buried with Christ in his death. By it we share in his resurrection. Through it we are reborn by the Holy Spirit. Therefore, in joyful obedience to your Son, we baptize into his fellowship those who come to him in faith. Now sanctify this water that, by the power of your Holy Spirit, they may be cleansed from sin and born again. Renewed in your image, may they walk by the light of faith
and continue for ever in the risen life of Jesus Christ our Lord; to whom with you and the Holy Spirit be all honour and glory, now and for ever.
So although it is true that one of the alternative "accessible language" Prayers Over the Water does not specifically mention blessing the water the doctrine of the Church expressed in her definitive liturgy, in the Alternative Service Book, and explicitly in two of the three Common Worship Prayers Over the Water does involve a priestly blessing of the water.
It is also clear though that, in common with the Roman Catholic Church, the Church of England believes nothing essential happens in the blessing of the water. Lay Baptism, under any circumstances, is accepted as valid by the Church of England, although it may be forbidden except in an emergency. This doctrine was somewhat disputed in the early nineteenth century and was the subject of a famous court case, Mastin v Escott in the 1840s, in which the matter was determined by the Judicial Committee of the Privy Council. (The matter, of course, was what was the doctrine of the Church, not what was truth - something this site would approve! The case involved a vicar refusing to bury the body of someone whose baptism he did not accept at a time when only baptised persons could be buried in the main churchyard.)
Baptism also differs from the other great sacrament in that priestly or ministerial consecration of the bread and wine is seen as essential.
In conclusion, there are strong arguments that Church of England baptisms do involve priestly blessings, but that they are no way essential.
Historically the Church has tried, up to a point anyway, to embrace ambiguity in one liturgy but increasingly offers alternatives which, as @DJClayworth points out, allow individual ministers to tailor services to reflect their personal viewpoints.
