Upon rereading what you are saying, I realize I am not disagreeing with you. I am, however, easily triggered by this as an armchair student of math, because I t

Upon rereading what you are saying, I realize I am not disagreeing with you. I am, however, easily triggered by this as an armchair student of math, because I think infinity forms a kind of a trap for amateurs like me. "Actual" infinity is not a member of the reals and it is not necessary to augment the reals to have a functioning real analysis. But you are right that it can be done, it can be added to the reals to form interesting algebraic structures and there are even books on nonstandard analysis that start from here (Elementary Calculus, which I have read part of but not finished).