By design, the answer is no. Toposes are models of intuitionistic bounded ZF. However, in general dependent type theories, including observational type theory, don't have (and don't want!) the powerset axiom.
There has been some recentish work on predicative analogues of toposes, but I'm not sure anything really definitive has emerged. However, I'm not super familiar with this area.
> To me, it still doesn't answer why TT is the most convenient axiomatization though. The internal language. [...] Especially because TT was arrived at from the IMO wrong angle, namely by coming up with a syntax first and its meaning second.
This is a very common misconception about what is going on when doing type theory.
From the point of view of a category theorist, the way to understand type theory is that it is a methodology for giving explicit presentations of free structures, in such a way that the morphisms make no explicit reference to composition.
The basic method of categorical logic is to compute a result in a syntax corresponding to a free structure, and then sending it to whichever non-free structure you like by the universal property of that structure. Most people initially find this a bizarre thing to do, since intuitively it seems like any proof that works in a free structure should work in any structure.
Indeed, this is actually true, if we only use the free gadget's universal property. However, if we have a type-theoretic presentation of a free structure, we actually have more information than the universal property: we know properties of the syntax! So we can exploit these properties of the syntax to establish facts which hold everywhere, by means of proofs that wouldn't work in an arbitrary structure.
For example, this is why type theorists devote so much effort on properties like cut-elimination: it ensures the composition of morphisms is automatically unital and associative.
Mike Shulman has written a fantastic set of notes, "Categorical Logic from a Categorical Point of View", which is one of the best explanation of type theory for "normal" mathematicians I've seen.