What I meant by "not a general past marker" is that you can't in general use positive 有 where you can use negative 没（有） to talk about the past. For example, you wouldn't say 我有看这本书。 to express you've read that book, but instead 我看了这本书。 Even in constructions like 有……过 the 有 is optional, so I'd say the past aspect is carried by 过.

On the other hand, maybe that is a shortening of an originally obligatory two-part expression similar to Jespersen's cycle, famously observed in the appearance of "pas" and ongoing disappearance of "ne" in French "ne ... pas". https://en.wikipedia.org/wiki/Jespersen%27s_Cycle