I have two baritone guitars; one is the Danelectro bari with 3 lipstick pickups and a "varitone" switch, and the other is an Alvarez Yairi acoustic.
The baritone, whether electric or acoustic - and they are VERY different from each other in sound and application - will never be the only guitar you play. In other words, it's extremely unlikely that anyone can say "I'm a baritone guitarist."
You pick one up from time to time because it will be effective for the song. If you need a different sound for a lead (Glen Campbell's "Galveston" comes to mind), or you need that tic-tac thing like early 60s Nashville ballads had.
You don't play a lot of full chords on it, because it takes up so much sonic space needed for other instruments and will tend to muddy things up. Plus, full chords are harder to play because the strings are thick and the scale is long.
It's a handy tool to have in your toolbox, but I don't think it's anyone's main tool.