We generally take it for granted that sentence 2 (because it is false) is not a theorem (and therefore sentence 1 is false), and the argument is meant to show that sentence 3 is therefore also not a theorem (even though it is true, since sentence 1 is false). This is a problem because we would like to use reasoning along the lines of sentence 3.

To see the real issue, you should replace sentence 2 with a conjecture whose truth you don't yet know but would like to know (and modify sentences 1 and 3 correspondingly). Now wouldn't you like sentence 3 to be a true? If you knew that sentence 1 was true, wouldn't you like to conclude that sentence 2 is true? Yet if you're PA, then you can't do that.