What options do I have for routers that are under $100 USD?

A "free" option if you have some PC hardware lying around is to run OPNsense/pfSense instead of buying a dedicated box. I say "free" since it will use more power and require more time to configure and manage.

Alternnatively, I ran one of TP-Links AX consumer router/AP combos for several years and it was solid. Even had an OpenVPN server built in. I can look up the exact model if you're interested.