As stated in the title, why do I need remote access to use Plex Amp when I have local access to the Plex Server on my local network? I don’t need remote access for the Plex App nor the Plex Dash, but why is it needed for Plex Amp?
You don’t. Read over https://support.plex.tv/articles/plexamp-network-connectivity-tips/ and make sure e.g. you don’t have DNS rebind enabled on your network for the Plex domains.