I think the .999 repeating thing has more to do with how we teach limits, people struggle to understand how two things that look definitionally different are the same because the rationale isn't well motivated. I had a professor that framed the whole process as an argument which was much easier to grasp, and I can easily imagine most people just getting a lecture or two on the equations and the implications.
Plus phrasing it as 'equals' or 'are the same' sort of puts the challenging part right in the front. Focusing on the difference more naturally leads you to the same conclusion.
It isn't a limit thing, though? It is a textual representation thing. You are not confused that 01 is the same as 1. Similarly, you are ok that .33... is the same as .3... In fractions, you are ok with the idea that 1/2 is the same as 2/4. And that you are often supposed to work in reduced form, such that you will probably never put 2/4 down as an answer to any problem. We work with "regular" numbers for so much of our life that it is odd to consider how to write them to that level.
Its not a textual representation thing, there are other repeating sequences that do not converge.
I mention limits because that was the context in which I learned about this, others have pointed out you can demonstrate this with just algebra (which makes me wonder if I just wasn't paying attention when that lesson came up).
Sure, but if your "basic algebra" includes the popular inconsistent axiom that "numbers have a unique decimal representation (up to leading and trailing zeros)" you can equally well just disprove any of the rules of algebra that you used to reach the conclusion.
When people have (accidentally) been taught that, and then they're presented with a case where it's false, it's perfectly reasonable that they wonder whether it's that thing that they were taught, or something else that they were taught, that is wrong. You don't have to look very far to find cases where it was the other laws of algebra that they believed were wrong, for example the popular proof that 1 + 2 + 3 + ... = -1/12.
Even a mathematically sophisticated alien who comes from a culture that somehow never used the rationals or reals the way we do might at first think that this is a fact that leads to a proof that no object that obeys your rules of algebra exists, rather than a proof that 0.999... = 1.
Most people, even most university students, don't have the benefit of a formal mathematical education that actually clarifies these kinds of riddles.
Plus phrasing it as 'equals' or 'are the same' sort of puts the challenging part right in the front. Focusing on the difference more naturally leads you to the same conclusion.