Let me clarify: if we interpret invariants as functors, then complete invariants correspond to conservative functors not faithful ones. I suspect that there isn't any conservative functor from either $\mathsf{Top}$ or $\mathsf{HoTop}$ to $\mathsf{Set}$, but this is not what Freyd's theorem is about.

I don't understand how this counts as an answer to the question as posed. After all, the question was about homeomorphism invariants and the category $\mathsf{Top}$ does admit a faithful functor to $\mathsf{Set}$, namely the standard forgetful functor. How is non-existence of such functor for the homotopy category relevant?

On reflection, I agree that the localization of a topological category should be defined exactly as you described. I would still worry about its homotopical meaningfulness though. The simplicial counterpart of the colimit you mention is always a homotopy colimit, but I wouldn't expect it to be the case in the topological setting.

Perhaps I should add that my answer is really about non-reduced homology and I'm secretly using relative homology all the time, but then the conclusion translates to reduced homology theories as usual.

Also, my answer is not "model-category-heavy". You do not need to know anything about model categories to make sense of this. I will try to elaborate to make this clear.

Well, in the times of Adams things like that were not sorted out yet. Why Hatcher says that this axiom is stronger (or if he actually claims that it is strictly stronger), I cannot say. But in the recent years, abstract homotopy theory (including Goodwillie calculus which puts homology theories in an abstract context) progressed to a point that things like that became folklore.

Yes, my statement is rather abstract, but any CW-complex is the homotopy colimit of its finite subcomplexes so the "direct limit axiom" follows. If you have any specific questions, I can try to elaborate.