Not recommending against RustDesk - it is a very cool project - but regarding the “Why?”, you could use a VPN or something like Tailscale which has MagicDNS that’ll resolve hostnames of computers to their local IP address. You can use this with GNOME’s RDP server to remote in from another device pretty easily.
Regarding your mouse double clicking, some Reddit users suggest pressing the click button quite hard a few times…