Let’s call the whole thing off. One might say the enemy army was “routed,” but do we ever use the word “routing” in that sense? It sounds wrong to me — does it only appear as a participle?
In the networking context though, people say “routing” either to rhyme with “pouting” or with “tooting.” I’d use the latter for “route 66” but I usually use the former for networking.
In case these linguistic musings bore people, fear not — I will write about other things soon.