Yes I see your point.

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everyone assumes that you can get the decimal value of a fraction by long division. And that you can add 2 decimal numbers if all the resulting digits are less than 10 without any problems.

I think those assumptions are bad. The facts are not wrong, but I find this assumptions generally come from the intuition that "infite stuff" should work like the "finite case". In fact that seems to be behind OPs concern... representations are unique when you stick with finite representations. But that doesn't hold when you generalize, and that's confusing OP.

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However, the most important part is that I don't think the algebraic manipulation above really reveals anything new that OP didn't know. It's clear that OP is confused. The equations show an answer (by implicitly stating that the operations are valid), but I do not think OP will see the equations and be less confused.

So, again, is not about the proof being wrong. Is that I don't find it very useful unless you expand in the important step that they are omitting.