I like your levels classification. I would say that for most areas of math, my goals fall closer to your "Level 2" than anything else. I don't want to be a mathematician, but I need to understand certain areas of math well enough to read papers / books that use that area of math. For example, ML papers that use calculus, probability, linear algebra, convex optimization, etc.
For what my goals are, I don't necessarily need to get to level 3 (although I might like to if I had infinite time available to indulge in every hobby/interest that I have).
Anyway, my feeling is that exercise solutions are very useful up to, or just around, what you call "level 2." And I expect that a lot of other HN users making this same complaint about solutions, are probably also working at or near that same level. For getting to level 3, I would agree with you that solutions are less useful (although even there I expect they would have some value, at least for self-learners).
It really is a different thing when you're trying to learn this stuff on your own, instead of being in the context of a classroom where you have an instructor and/or TA's that you can get to check your answers. For those of us in that mode, anything that adds friction to the process is extremely frustrating, at least from my subjective perspective.
This may contradict what you have been told—that mathematics is sequential, and that you must understand each sentence before going on to the next. In reality, although mathematical writing is necessarily sequential, mathematical understanding is not: you (and the experts) never understand perfectly up to some point and not at all beyond.
I expect this is another of those things where it kinda depends on where you are. Getting up to and through, say, Multi-Variable Calculus does seem to depend on a lot of earlier subjects that depend on earlier things, in a roughly sequential stream. But beyond that, I can see that things start to diverge and become not-strictly-sequential.