There are no abstract categories $\newcommand\Set{\mathbf{Set}}\newcommand\Ob{\mathbf{Ob}}\newcommand\Hom{\mathbf{Hom}}$Work in a foundation that admits a countable hierarchy of notions of ‘set’, and say that a category is $n$-small iff its object and arrow collections are $n$-sets. Denote the category of $n$-sets by $\Set_n$.
For each $n$-small category $\mathcal{C}$ and each object $X\in\Ob_\mathcal{C}$, define
\begin{gather*}
\Hom_\mathcal{C}(\ \ ,X)=\operatorname{cod}^{-1}(X)\subseteq\Hom_\mathcal{C} \\
\Bigl({}=\bigcup_{W\in\Ob_\mathcal{C}}\Hom_\mathcal{C}(W,X)\Bigr).
\end{gather*}
For each $n$-small category $\mathcal{C}$, define a functor
\begin{gather*}
\Hom_\mathcal{C}(\ \ ,-):\mathcal{C}\to\Set_{n+1} \\
X\mapsto\Hom_\mathcal{C}(\ \ ,X) \\
f:X\to Y\longmapsto {f\circ{}}:\Hom_\mathcal{C}(\ \ ,X)\to\Hom_\mathcal{C}(\ \ ,Y).
\end{gather*}
$\Hom_\mathcal{C}(\ \ ,-)$ is always faithful, so $n$-small categories are never abstract. Working in an appropriate foundation (see An axiomatic approach to higher order set theory (disclaimer: I am the author of this paper)) all classically considered ‘abstract’ categories like Freyd's homotopy category are just $0$-large $1$-small categories, admitting canonical faithful functors into $\Set_2$.
Taking this view, it seems like categories are only ‘abstract’ if we work in a set-theoretical foundation that is ‘too small’ to see the larger categories of sets that all categories embed into. This makes me wonder,

Are there any categories we care about that remain abstract in set theories admitting a countable hierarchy of notions of ‘set’? What if we extend the hierarchy of notions of set to arbitrary ordinal heights?

 A: The argument you give in your original post is essentially the proof that every small category is concrete.
So if you work in a setting that have enough universes/inaccessible cardinal/notion of smallness so that it is reasonable to only consider "small" categories, then of course all categories are concrete.
However, as soon as you allow yourself to form something like the category of all sets, you will get examples of non-concrete categories:
The typical examples of non-concrete categories are the category of all groupoids with isomorphism classes of functors between them. Or the homotopy category of all spaces/simplicial sets.
So unless you insist on only talking about small categories (which definitely make sense in some foundation), you'll always get examples of non-concrete categories.
But I would add that even if you consider that "all categories are small", a better point of view on concreteness is that, (like for smallness) one shouldn't talk about "concrete categories" in the abstract but of $n$-concrete categories exactly as you talked about $n$-small sets. Even in settings where everything is small in some sense, the notion of smallness is still relevant: for e.g. the category of $n$-small sets isn't closed under all $m$-small colimits for $m>n$. So even in a setting where every category is concrete in some sense, the notion of concreteness could still be relevant.
