Why is it matching the final " if it could leave it for "? and the pattern would still match?

In the greedy case, it's because that's how regexes work; if .* can match the quote ("), it will.  Note that it would be different if the trailing quote was non-optional; then the greedy match would only match up to but not including the final quote.

In the non-greedy case, of course, it's the same problem; the non-greedy succeeds by matching nothing at all, as the optional final quote does not have to be matched either.

Here's another way to look at the effect of a "non-optional" final quote for both the greedy and non-greedy cases:

    $string =~ s/\"(.*)\"/$1/;
    #                  <= Final quote " pushes left against
    #                     otherwise greedy match (.*)

    $string =~ s/\"(.*?)\"/$1/;
    #                   => Final quote " pulls right against
    #                      otherwise non-greedy match (.*?)
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So when the final quote is optional, neither of the above constraints get enforced; the greedy match can be maximally greedy, and the non-greedy match can be minimally greedy.

And by the way, you don't need to escape the quotes in a regex.  \" can be just ".