Why not use large $k$ in a $k$-ary tree?
BTrees are used in practice - file systems, database with $k$ for example equal 1024 or 4096, so it seems to be bigger than binary.

Probably you have not encountered need yet.
For example ternary tree with comparing >, =, &lt; gives theoretical improve $O(log_3 n)$, but self balancing trees in basic form do not accept duplicates, so there is more overhead and not strictly practical.
But bigger ones, yes, they give boost and you use it right now (on the computer).