Why is /etc/profile not being loaded during non-login bash shell sessions?

Per default, gnome-terminal does not start bash as a login shell (I assume you mean bash started inside a gnome-terminal). That means bash will not read /etc/profile or ~/.profile. As you have correctly observed, bash will read those files if started as a login shell.

The quick fix to your problem is to configure gnome-terminal to start bash as a login shell so it will read /etc/profile. To do that you have to enable "Run Command as a login shell" in the gnome-terminal "Profile Preferences" reachable from the Edit menu.

I do not recommend doing this because it messes up the distinction between ~/.profile and ~/.bashrc. Ideally ~/.profile should do stuff which is required only once, during login, while ~/.bashrc should do stuff which is required everytime bash is started.

There is a question and answer at superuser talking about the difference between bashrc and profile. Read there for more information.

From your problem description it seems that the rvm script needs to be loaded only once, during login. As far as I know Ubuntu has configured the graphical login to read /etc/profile/ and ~/.profile. That means, after logging out and a logging in once, the rvm script should be active. If it still doesn't work, then perhaps the rvm script needs to be loaded for every bash session. If that is the case then bashrc is the more appropriate place for the script.

There is, however, a file /etc/bash.bashrc that is read by gnome-terminal and is the "system-wide .bashrc file for interactive bash(1) shells."

My call to the rvm function, [[ -s "$HOME/.rvm/scripts/rvm" ]] && . "$HOME/.rvm/scripts/rvm" , went in there, and works just fine for the couple of users on that system.

Installing RVM as multiuser requires the user to run this command:

(because Ubuntu does not source /etc/profile.d at the login)

echo '[[ -s "/etc/profile.d/rvm.sh" ]] && . "/etc/profile.d/rvm.sh" # Load RVM function' >> ~/.bashrc