I believe the etymology of 'router' (sending network packets along a route) and 'router' (cutting wood) are different. They are pronounced differently in British English, at least.

Rout, via Old English 'wrotan'

https://www.etymonline.com/word/router

Route, via Old French 'rute'

https://www.etymonline.com/word/route

Logically, one cannot create routes without a router and correspondingly, a router bit ;)

Also 'router table' vs 'routing table'.