Is $LL(k)$ for large $k$ considered harmful? If so, why?
The parsing table of a $LL(k)$ grammar grows exponentially in $k$.
This is however the worst-case scenario, which is not typical, as Niel pointed out:
For an $LL(k)$ grammar $G$, $k$ is the size of the maximal lookahead needed across $L(G)$.
But it appears that in practice, the need for a $k$ lookahead is restricted to a small portion to the language, so that the size of the parsing table remains reasonable.

We face here a quite common problem of theoretical complexity not reflecting the practical experience.
Beyond $LL(k)$ are $LL(*)$ grammars, whose lookahead is adaptative, so as to keep the parsing table no larger than necessary.
$LL(*)$ technique is examplified by ANTLR, and described in this paper by Terence Parr.

Finally, I would say that many teachers focus on $LL(1)$ to emphasize that $LL(1)$ suffices to parse most languages in practice.
Philosophically, our programming languages lie somewhere between Deterministic and Non-Deterministic Pushdown Automata, but in fact way closer to DPA than to the full NPA.
A similar observation can be done for natural languages: context-free is not enough, yet context-sensitive is way too powerful.