The universe as we know is quantised, it's finite and any process can't be smaller than Planck length. That means that within a brain, there are finite number of possible combination of matter, which leads to inevitable loop somewhere along the way, assuming that's deterministic. Even if it is undeterministic and the chains of events are not repeating, your sets of possible memories are finite, so at least the things that you can remember will start repeating. That's what I mean by "traditional sense", your life goes on as it is now.
Of course, your brain can expand during your immortality, grow larger, you can become the Face of Boe first, then slowly consume the entire universe, and then you can use your perpetual motion machine to expand it even further, or you can use your Planck divisors to make your post-neurons smaller and smaller. So, yeah, immortality can make sense if everything grows in complexity in that manner. I would question if all of these things happening at once are more likely than another universe that was more complex to begin discovering your thoughts inside a simulation that happened to touch on our universe, which would still go to my point that there are substitutes for immortality that might be just as good without having to make you immortal.
As for your other point – I didn't mean the ability to verify things personally within your lifetime, that would be insane, the universe is so complex that you can't understand more than a tiny portion of it even if you're a genius, let alone test everything yourself.* I meant that you can't use the scientific method to reason about things that we can't observe at all, including your own personal quantum afterlife.
*Well, that might be an exaggeration, there are simplified ways to learn and test that don't get into unnecessary detail.