Obviously binary trees are great because of $O(\log_2 n)$ search, inserts, and deletes in best case.
To "maximize" occurrence of best case, we can use self-balancing trees like red-black trees, AVLs, splay trees, etc.
If we use a $k$-ary tree, we can get $O(\log_k n)$ searches, inserts, and deletes in best case. Similarly we can "maximize" the occurrences by using self-balancing trees.
I know these are asymptotically of the same order, i.e. $O(\log n)$, but the constant could be huge in practice.
The huge reason I can think of is that by using larger $k$'s, you lose the ability to make a decision based on one comparison.
What are other reasons larger $k$ $k$-ary trees aren't used in practice?