I actually find the appearance of the former example less ugly. Probably because I’m not an American? Though I agree with your point.
Alternative plan: the quotation marks go directly above the period, in ALL cases. Anarchist route!
The ambiguity it’d introduce would be pretty minimal. Also, if all our phones suddenly started doing it, I’d give us about 20 years before nobody younger than 15 knows there was ever a rule to begin with.
Only problem is that you’d still need to know the rule for all other end punctuation… hmm…
You know what? English doesn’t make sense anyway, it’s fine.