The basic “equation” for anicca seems to be something like: impermanence+attachment = dukkha. That seems fairly convincing and uncontroversial to me (well, assuming dukkha implies ups and downs, joys and sorrows, not just uniform bleakness).
This then becomes a problem to be solved (a solution being some notion of salvation). A solution can involve tackling the attachment, focusing in on the impermanence bit, or perhaps a mix of both (most approaches usually involving a few more metaphysical assumptions).
Brahma/Atman-type systems target the impermanence bit, asserting that actually really there is a permanent part immune from this problem; though they target attachment too somewhat, some kind of spiritual development via reincarnation where attachment is reduced until one reaches the Godhead eventually.
YOLO (You’ve One Life Only) Buddhism assumes even less. I think this mostly just works with the above basic equation, makes no assumptions about past or future lives or karma, and targets reducing attachment for a life with less dukkha now. And I guess there can be YOLO notions of self/atta and anatta too. In this case, I guess the person has to assume this will be worth the effort (perhaps otherwise going more towards a “eat, drink and be merry for tomorrow we die” type philosophy 
). Ancient Greek Epicurianism seemed to be somewhat in the middle (advocating living quite a simple “good life”).
Early Buddhism does, of course, have its own metaphysical premises. A core one is that there is nothing permanent anywhere to be found. Effort then obviously has to be devoted to tackling the attachment side of the equation to reduce dukkha (there’s no other option left).
This then is where atta and anatta comes into things. I guess for the YOLO case we can have a self as a kind of ephemeral physical/biological operating system for the body/mind (rather like a mobile phone OS; it exists in a sense but vanishes when we turn the phone off or it is destroyed, and there being no continuity to/from any other future/past mobile phones ). It’s there but completely transient. I don’t think it’s invalid to call this a “self” even it is not the “self” of other systems.
Buddhism asserts quite a bit more about such a “self”, that there’s kind of causal link between lives (and associated karma also). It’s still rather ephemeral but in a more persistent life-spanning way. Actions have consequences. “Eat, drink and be merry” now has more definite downsides. Salvation and the solution to attachment is now is placed in a much larger context.
More assumptions are being made about a “self” than the materialist or YOLO approach. However, it doesn’t come with all the Atman baggage (though there is still rebirth and karma). It is still impermanent and still is in a sense just a chain of cause and effect. It’s assuming more than the minimalist YOLO but less than the more maximalist Brahma/Atman systems.
I think the Buddha’s argument that if there really was an imperishable atta at the core of your being, then the atta should simply be able order its various aggregates to be such and such (choose sukkha over dukkha whenever it wanted) is a good argument. It’s a reasonable question to ask “atta” believers. There are always workarounds of varying credibility to such things though. A believer could argue that a spiritually advanced enough soul (“ascended master” or Taoist immortal) could, of course, do such a thing. Conveniently explains why the believer themselves can’t do this right now!
I think one has to try to reverse-engineer from the suttas to try to find a consistent meaning for atta (and therefore what anatta is denying). Is it just “self” with its usual connotations in the English language? I’m not so sure. There’s certainly a lot more metaphysical baggage for the dukkha to anatta leap, I think, in comparison to going from anicca to dukkha (which is more like a more straightforward causal jump). Buddhism is trying to put clear water between itself and both the Atman/Brahma and YOLO/materialist approaches. Anatta seems to me to be more about making clear water with the Brahma/Atman systems. Concepts like kamma and dependent origination are more about clear water with materialism/YOLO approaches. Kamma, DO and anatta, I think, are all metaphysical jumps beyond the basic anicca proposition.
I don’t think you can really determine the veracity of such things by argumentation. Inherently these are just different metaphysical assumptions about the universe built on top of the basic anicca+attachment=dukkha proposition, assumptions which may or may not be true.
