Aim Assist falls into a category of cheats that are more or less undetectable and unavoidable over the internet: skill assists for something a computer does better than a human. How central these are to the game depends on the game. For a game like Chess, the impact (of consulting a computer to suggest moves) is devastating, but the online community survives. I think it's typical in such communities for truly high stakes competition to happen in person, and for the online scene to be seen as more of a social / practice scene. I like this solution: prevent theft by reducing the value of what can be stolen.
Games that turn heavily on aiming have a similar central security flaw in that it is hard to prevent cheating at the game's central skill. (Though I think in the case of aimbots, sometimes webcams are substituted for LANs, with some success.)
On the other hand, some games are practically cheat-proof. A puzzle game in which you submit actual solutions doesn't require any trust of the client at all. CTF games generally run along these rules - almost anything you can do to solve the puzzle (googling, teaming up, writing tools, bringing AI assistants) is considered fair game. What might be considered a cheat in another context is just advancing the state of the art.
HUD improvements depend on the game. But as a simple example, I play a game where leading a moving target is a major skill; a HUD that gave you an aimpoint for a perfect intercept would be a pretty big cheat.
I think anti-cheat is one of those problem spaces where there is a danger of overemphasizing technical solutions to social problems. Technical solutions are nice, but there are also gaming experiences that are only practical on a private server, with friends, on the honor system. A wise friend once observed that removing griefers and jerks from a community also did a lot to address cheating. I think it is best thought of as a social problem first, though I agree it all depends on the context.