Allowing the quote to be affected by the punctuation around it seems to undermine the “verbatim”-ness of a quote. If the period goes outside of the quote, then the quote is always a discrete unit of text that can be moved around the sentence as needed.

Example:

He said, “It’s fine”.

“It’s fine”, he said.

I would accept always including the period inside the quote for that case, but it causes other problems. If you put the period inside the quote, how do you indicate a quote that must end in a period, but does not end the sentence?

Example:

The spec sheet said “88 m.p.h.” on the back.