To be sure, the episode doesn't really state that aging involves the alteration of the DNA. Rather, it states that an alteration of DNA brought about the aging-like effects in Pulaski, which is a very different thing.
So the heroes knew for sure that her DNA had been altered, and that this was the cause of the eerily aging-like woes. They could only hope that removal of the cause would restore the original state. And for that purpose, cross-comparison between the current Pulaski and a sample of her former DNA would be very relevant.
This is made clearer in the penultimate script (see TrekCore) than in the aired version, but the story logic still works for the aired version as well.
So no instant youth. Just instant de-wrinkling and de-arthritising, things that Pulaski or even Crusher can do at their leisure without the transporter. (The transporter was only needed to stop the process that was causing the pseudo-aging in Pulaski, and only incidentally used for reversing the symptoms after that.) But at some point, age still gets you, even if you happen to be de-wrinkled. So most 24th century people don't bother with such treatments.
Howzzat sound?
Timo Saloniemi