Can I make character movement entirely with Mecanim's root motion?

Basically, yes, this is possible. You would most likely need to create two movement state machines, one for climbing and one for regular locomotion. The movement itself would then be applied via root motion (if you set it up correctly, that is). "Just setting parameters for the animator", however, isn't as trivial as you make it sound, especially climbing does require quite a bit of effort to get right when to feed which parameter to the animator.

So the rigidbody isn't necessary unless I do some kind of ragdoll at some point?

As for the climbing, I'm only gonna add ledge grab and climb up while hanging on the ledge... Not necessarily ladders or anything like that.

I just fiddled around and noticed collisions would be a porblem. How do I make the player respect physics without gravity being applied?

For the ledges I use colliders ongameobjects with tag "ledge" to trigger grab animation. Only problem so far is y-axis motion up and down. I have falling animation that has the root motion for falling down and a jumping animation that has root motion for up. gravity makes my jumps fall short and falls stretch too long.

You'd need a character controller so gravity would have an effect. This also conveniently lets you check if the character is grounded, allowing you to play a fall animation if it's not.