I don’t think I have enough knowledge to solve this or say anything for certain, but I wonder if the power button is treated as an external keyboard and is getting ignored in tablet mode?

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.

PairDrop(dot)net is a fork with a bit more features

Strange. The default keyboard works wonders for me but doesn’t automatically pop up. Are you able to swipe from the bottom edge of the display to make it pop up?