The session/window/pane/instance thing is definitely confusing and not well explained. I think it's even worse if you already have a lot of screen experience.
screen has the equivalent to sessions as well `screen -ls` will show them whereas `tmux list-sessions` will show the tmux ones. A window is composed of panes (analogy from physical windows, pane of glass). An instance is one user on a session.
Can you name one distro that doesn't include it by default? In my experience you can rely on just about every system having a core set of utilities: bash, vim, screen, ssh etc.
But if you have your own workstation and do most of the work with this workstation, tmux + customized configs might be a viable option. It just depends how good your muscle memory will keep up. :)