If the Bard wants to be pedantic he should at least get his facts straight. The 'who' and 'that' that you talk about are called relative pronouns and according to Michael Swan, either is fine; eg. 'There's the man who/that sold me my bike.'
Secondly, having 'rewritten' the original he completely lost the sense of the statement, which is that 'would have' is wrongly written as 'would of' because in its contracted form it sounds the same as 'would have'.
This is great fun, thanks! ps. Michael Swan is the absolute guru regarding English grammar and usage!