Nice! Thanks for those detailed screenshots!
I think that those files/folders being owned by root are the issue. You should be able to fix those permissions by deleting that folder and then letting Local re-create it. To do this in the terminal (likely the easiest way to do it) start with quitting Local and opening terminal.
Within the terminal, run this command to delete that router configuration folder:
sudo rm -r ~/Library/Application\ Support/Local/run/router
This will prompt you for your MacOS user’s password since you’re trying to delete files owned by root. Once that’s done, start Local and start a site which should re-create that router configuration folder. Let me know how that goes, or if you run into any other issues!