Your response didn't answer that quote. The giving away comes before the vows so they don't really have any correlation if that line comes after, unless I am missing something.
The answer is no. It's been optional for many years.
According to the Church of England website, at the bottom of the page (Notes on to the Marriage Service) it says (asterixes are mine)
"6 ‘Giving Away’
This traditional ceremony is optional. Immediately before the couple exchange vows (here here), the minister *may* ask:
Who brings this woman to be married to this man?
The bride’s father (or mother, or another member of her family or a friend representing the family) gives the bride’s right hand to the minister who puts it in the bridegroom’s right hand. *Alternatively, after the bride and bridegroom have made their Declarations*, the minister *may* ask the parents of bride and bridegroom in these or similar words*:
N and N have declared their intention towards each other.
As their parents, will you now entrust your son and daughter to one another as they come to be married?
Both sets of parents respond:
We will."
(This is what I remembered, a form of the line I quoted did exist in the earlier services but like the "who gives...?", it is also optional).
https://www.churchofengland.org/pra...p-texts-and-resources/common-worship/marriage
There are 3 approved forms of words for the marriage service in the Church of England:
The Book of Common Prayer (the oldest - 1662), Alternative Services: Series One (which William and Catherine used) and this one, formally called "The Marriage Service from Common Worship" aka "Modern Rite". It was formally approved in the Church of England in 2000 (although I used it in 1985 in the Canadian Anglican church.) A couple may select any of these 3 rites, it's their choice. Holy Communion is also optional.
BTW, Anglicans/Church of England/Episcopalians have never said "I do". The correct phrase has always been "I will."