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I agree with the update to the answer. Note also that in the rest of the talk, I was doing the same switch of notion related to what a 6FF is. (Somewhere I mumble something about the functor from Corr(Sch) not being part of the data in the construction of the (\infty,2)-category of motives.)
Sorry, I did mean to cite Drew-Gallauer, but more in spirit than in details. When giving my course, I did convince myself that one can prove a precise initiality statement. The claim in the talk was meant to be slightly imprecise as during the discussion I was slightly shifting the intended meaning of 6-functor formalism; in particular, when working with the 2-category, I do not want to enforce a priori a functor from the correspondence category. (Relatedly, a uniqueness conjecture I made in my lecture notes is very likely false. Maybe this is why you are skeptical?)
(In any case, either take $\kappa=\omega_1$, or take a colimit over all $\kappa$ in the end; both ways lead to a theory of presentable $(\infty,n)$-categories that is independent of choices of universes.)
(One can replace $\omega_1$ here by any regular cardinal $\kappa$. One might object that I'm secretly introducing universes by this choice; I would argue against it. In particular, for any fixed $\kappa$ the above scheme can be used to inductively define $\kappa$-presentable $(\infty,n)$-categories for all $n$; no need for a chain of $n$ universes. Also, I have the feeling that it may actually be helpful to fix $\kappa=\omega_1$, similar to how in condensed math we recently switched to the light setting.)
Unfortunately, my list of things to write is way too long. Short: As I wrote somewhere on MO, I like the notion of presentable $\infty$-category. In practice, all of them are $\omega_1$-compactly generated, and functors preserve $\omega_1$-compact objects. Such form an $\infty$-category $\mathrm{Pr}^{\omega_1}$ that happens itself to be $\omega_1$-compactly generated presentable, so $\mathrm{Pr}^{\omega_1}\in \mathrm{Pr}^{\omega_1}$. Presentable $(\infty,2)$-categories can then be defined as $\mathrm{Pr}^{\omega_1}$-modules in $\mathrm{Pr}^{\omega_1}$, etc. (I learned this from Ko Aoki.)
@CameronZwarich I claim that carefully thinking about universe issues may make it possible to eliminate them, but it requires one to think very carefully how to formulate things. (Again, staying in ZFC is not my actual concern, but a likely outcome of such a careful reflection.) I've recently thought more seriously about higher categories (in particular, categories of categories) and finally have a way of thinking about them that I like. Notably, this way does not involve a choice of universes (and instead features a form of $V\in V$ that happens to be literally true).
Whoops, you are right. (For the application, it only matters to have this limit property on semi-simplicial sets, and that each term admits some refinement to have a simplicial set.) But you can correct this by approximating also the degeneracies mapping towards $T_n$ (so $T_n$ and all $T_{n,i}$ come sandwiched between two given things).
