> There are legit reasons to have a router be publicly accessible.
No, there are not.
> How else would one remotely manage a router
Over a WireGuard connection to a secure management network.
> The only real issue would be using a default password
Uh, no. Try any number of CVEs or 0-days or unknown-until-it's too-late vulnerabilities, depending on what web daemon/frameworks are used by the router's management software.