@KevinCarlson I also just realized a further problem that it's bot clear how to get the correct 2-morphisms i.e. the natural transformations out of this.

@CharlesRezk Actually now when I think about this why don't I just take the 2-category of topological categories and localize w.r.t. the essentially surjective functors inducing weak equivalences on all mapping spaces. This might already give me the homotopy 2-category of $\infty$ categories and it seems like the simplest and most direct way of combining ordinary category theory and the homotopy hypothesis. I'm not sure why I didn't notice this before...

@CharlesRezk If I had to guess I would bet not. What I do think may be possible (probably being naive here) is to get homotopy category of $\infty$-categories by melding together in a clever way the category of topological spaces with their weak equivalences and the 2-catgory of 1-categories. We already know it can be done in a myriad of different ways many of which give equivalent resulting 2-category. The question is whether we can do this using only 2-catgorical arguments, or rather whether the construction has a certain 2-categorical universal property if you like.

@CharlesRezk I'm sorry you had to read through all that just because I was afraid of being shut down. I hope future readers will not have to go through this travesty.

@CharlesRezk I completely agree with what you said. I was afraid if I would phrase the question as simply as I did in the title it would be dismissed as imprecise and voted to be closed. Let me add a remark at the start pointing that out.

@DylanWilson I was under the impression that in this paper one starts off with a working theory of $(\infty,1)$-categories then builds on that. Maybe I'm missing your point.