Snap-installed app doesn't put icon into launcher
I found the answer. It's not Plasma/KDE or snap, but zsh. I'm not sure why this worked on my old setup as I was using zsh there as well without a workaround.
This answer solved my problem: https://askubuntu.com/a/989485/754993.
To quote:
If you are using zsh, the snap binary and desktop directories will not automatically be added to your environment variables.
The step I did specifically was to add emulate sh -c 'source /etc/profile.d/apps-bin-path.sh' to the bottom of my /etc/zsh/zprofile file.
Solved it by adding "/snap/bin" path to /etc/environment file.