In this generality it is certainly false. Let $C$ be the coaugmented counital coalgebra over $\mathbb{Q}$ cogenerated by a single primitive cogenerator in (homological) degree $3$. Then $\pi_{\ast} (Comod_C(\mathbb{Q})) = \mathbb{Q}[x]$ with $|x| = 2$ so $\mathbb{Q}$ can't be compact because then $\mathbb{Q}[x^{-1}] = 0$ contradicting that $\pi_{\ast} (End_C(\mathbb{Q}))[x^{-1}] = \mathbb{Q}[x,x^{-1}] \ne 0$

To add to what Achim said, with our current technology (by which I mean HTT) there is no way to define the $\infty$ category of $\infty$ categories directly as a quasi-category (as far as I'm aware). Instead it is defined as a homotopy coherent nerve of a simplicial category (e.g. fibrant marked simplicial sets). So in HTT a functor into $Cat_{\infty}$ is encoded as a simplicially enriched functor. This forces us to leave the world of quasi-categories. To remedy this S/U allows us to talk about functors without leaving the world of quasi-categories by working with the unstraightned objects.

I believe this is a really standard example which is probably known to many but I only realized now. Probably you were already aware in which case I apologize for the cluttering this comment section

I think I have an example for failure when $\kappa = \omega$. If we take $C = \otimes^{\infty}_{k=0} E\{x_k\}$ a countable tensor product of simple coalgebras cogenerated by a single primitive element (over $\mathbb{F}_2$). Then in the $\infty$-category of $C$-comodules (in $\mathbb{F}_2$-complexes) there are no non-trivial compact objects. Essentially because for any compact comodule there's an non-nilpotent element in the Ext of the identity functor which acts on it's underlying complex by 0.

Oh you're right! I realized I was quoting the wrong thing. A page before there's exercise 2.j: "Prove that if $F_1, F_2: \mathcal{K} \to \mathcal{L}$ are $\lambda$-accessible functors between locally $\lambda$-presentable categories and $F_2$ preserves limits, then the inserter category $Ins(F_1, F_2)$ is locally $\lambda$-presentable." but as you say the category of $S$-coalgebras is $Ins(Id,S)$ and that's the wrong direction. The next exercise is the same only instead of $F_2$ preserving limits they require $F_1$ to preserve colimits and no bound on the cardinal of presentability :/

I just found the following exercise (page 127 ex. 2.l.) in Adámek-Rosicky's book: "Prove that a lax limit of locally $\lambda$-presentable categories and limit-preserving $\lambda$-accessible functors is locally $\lambda$-presentable". I believe the formation of the category of coalgebras over a comonad is a special case of a lax limit (a sort of lax equalizer). So at the very least even if this result is false (which now seems much less likely) a counterexample was not known to them when they wrote the book.

That's a good counterexample for the case of coalgebras over just endofunctors. But I must say I have little to no experience with this notion so I can't decide if it counts as positive or negative evidence towards the statement for Comonads... Which I would be delighted to have a proof/counterexample for.

Do you have a good example of a failure of this when $\kappa =\omega$?

@MikeShulman "... frequently turns out not to be the case in locally-presentable-category theory." What are some good examples demonstrating this?

@TomLeinster Thanks, I tried to fix the phrasing a bit to make it clearer.

It's perhaps not the most important reason but "complex oriented" beyond the existence of chern classes means you can "integrate" cohomology classes along submersions of almost complex manifolds. I believe this was a crucial part in Hirzebruch's original proof of HRR theorem.