Never mind. Seems to be an issue due to running Local inside Hyprland’s Systemd unit, so stopping the site stops Hyprland too. If anyone else has this issue on Hyprland, just run uwsm app -- local-by-flywheel to run local as a separate unit. See Systemd startup – Hyprland Wiki
Edit: To be able to launch Local from a launcher like wofi without any issues, check this: GitHub - Vladimir-csp/uwsm: Universal Wayland Session Manager