Thanks for this report, @michl!
It does seems specific to macOS 15.4 — we had an earlier report here:
Workarounds so far are the ones you mention: switch router mode to localhost in Local’s preferences, or update your /etc/hosts manually.
We have a ticket to look into this but don’t have a technical explanation without further research yet. (Local should open an auth dialog box after the “host redirection” message you mentioned to ask for your password to write to /etc/hosts. That prompt applet is provided by a dependency and isn’t launching under macOS 15.4 beta, which is what we plan to investigate.)