True, but what about the brain itself? Doesn't it deteriorate with age, too, just like any other organ? Don't know myself. Just asking. This stuff is too fascinating!
Imagine extending that scenario. You could stay healthy and young looking for virtually forever, but your brain goes senile. Imagine a geriatric ward where everyone is in great physical shape, everyone looks like gorgeous 20 somethings but all have Alzheimers.
That's it in a nutshell. If you can't prevent or treat dementia, the rest may be a Faustian bargain.
Deterioration aside, you'll eventually run into capacity issues. The brain forms new memories and learns by establishing new neural connections. While we don't know exactly what the brains capacity IS, there has been some research indicating that it wouldn't hold more than a couple hundred years worth of memories.
What happens when the brain fills up? I'm sure science will figure out a way to reset connections and free up space by then, but which memories will you be willing to lose forever? Theoretically, someone could end up living to be 500 or more and have no recollection of their childhood or college years.