Following the Debian discussion https://bugs.debian.org/cgi-bin/bugreport.cgi?bug=752129, Bug #794315 “lightdm forgets to source /etc/profile and ~/.profi...” : Bugs : lightdm package : Ubuntu, and https://bugs.debian.org/cgi-bin/bugreport.cgi?bug=752192, it appeared that lightdm does not and will not source /etc/profile and /etc/environment by default. User is expected to do ~/.xsessionrc. In my case, source via ~/.bashrc is making a lot more sense. Hence, based on https://superuser.com/questions/881102/etc-profile-not-being-sourced-at-login, we can use:
test -f /etc/environment && source /etc/environment
test -f /etc/profile && source /etc/profile
This restores the terminal based snap commands, since it’s lightdm specific matter.
As for .desktop, no update yet, still investigating.