Seventeenth century grammar and the grammar of today often differ, so I would disagree that they are the ultimate arbiters on questions of usage today, however impeccable their grasp of seventeenth century grammar.
On the issue raised, I tend towards your position with the example, but I also think that the other side has a case to make and it is definitely true that people say things like you that's, though probably as much for euphony as anything else. (Edit: Actually, after sleeping on it, the more I run the sentence round my head, the more logical Bhaissahab's position seems- it's not clearcut)
BTW The 1662 Book of Common Prayer doesn't use 'who', but 'Our Father, which art in heaven' : The Order for Evening Prayer > BCP, an interesting difference that possibly shows a different way of viewing the divine:
Our Father, who art in heaven
hallowed be thy Name,
thy kingdom come,
thy will be done,
on earth as it is in heaven.
Give us this day our daily bread
And forgive us our trespasses,
as we forgive those
who trespass against us.
And lead us not into temptation,
but deliver us from evil.
For thine is the kingdom,
and the power, and the glory,
for ever and ever.
1662 Book of Common Prayer
Our Father, which art in heaven, Hallowed be thy Name; Thy kingdom come; Thy will be done; in earth as it is in heaven. Give us this day our daily bread. And forgive us our trespasses, as we forgive them that trespass against us. And lead us not into temptation; But deliver us from evil. For thine is the kingdom, the power, and the glory. For ever and ever. Amen.
(I have italicised the differences and emboldened some of the semi-colons)