There are big and hairy problems that are bad investments for a young mathematician. I would steer students clear of the Collatz conjecture. But once you get up to speed in your research area, you usually find interesting problems thick on the ground.
Tenure-track positions are competitive, but I don't think there are a lack of interesting things to work on.
99.99% of everyone won't be remembered for their "contributions" so why not do something you enjoy?
Having said that, if something does come along that interests someone, they should try it. A friend of mine is doing his 2nd PhD 40 years after his first, having found his way into it via a love of jazz music.
It's not easy, but it's certainly very impactful.
Otherwise, I will try to come back to this comment in a few days. I'm a bit busy with an urgent deadline, but I'm glad to expand on this later on!
I wish I just had more time to do math like the OP but the saturation of the field especially with people who can deeply understand the abstraction is very very intimidating.
https://en.m.wikipedia.org/wiki/Evil_demon
(apart from saying that the evil demon is a future type computer)
The first incompleteness theorem says that for any consistent formal system T (with a recursively enumerable set of axioms) capable expressing of elementary arithmetic, T can express a statement which it can neither prove nor disprove.
The second incompleteness theorem says that T can't prove the statement "T is consistent". (I've still glossed over a number of technical details here; pick up a book on model theory if you want all the messy internals.)
First order logic is notably not capable of expressing elementary arithmetic. And observers aren't involved in any way.
Sorry if you can’t read deeply into it or something. I’m not posting for grad students. I can sense you just like to correct people. Ahhhh I’m so wrong, you’re right?
It's tempting to try to apply them (or rather the same kind of conclusions) in other (non math) contexts, but it's very not obvious that you'll get something sensible. While you can play with the ideas, invoking Godel's theorem outside of its specific context doesn't make much sense.
I guess you could call a Turing Machine implementing the search algorithm for proofs implied by a logic an "observer", since it produces "reachability observations" i.e. proofs.
As a field I think math is well past its diminishing returns imho. It's like 'what was the last big philosophical discovery'...yeah...hard to think of one. Maybe the P zombie concept or the simulation hypothesis. But new and important books, fiction and non fiction, are being written all the time.