Depends on the character background... As you stated the standard skill set places them in contact with all crew in stressful situations along with a diplomatic role that makes them excellent on all away teams. They are also present on the bridge a lot.
This puts them front and centre of all the required action hot spots. But you are right the standard definition doesn't leave much to do.
But.
A full blown Betazoid gets pretty decent sweeping psionics that would give them a massive edge over 'norms' and skills unavailable to others. This also includes hand to hand combat. Somewhere I stumbled over Betazoid Martial Art; Tassa’Akai including a hand to hand Baton (Akai) which I imagine like the Minbari fighting pike.
A counselor has no active duty station. But Starfleet encourages cross training. Perhaps the Counsellor would look into emergency back-up of a duty station (of a NPC) or managing emeregncy response (fire control and first aid).
Or there is the sciences. Another role that is oftenunder staffed in the game... Pick a field of study.
DanG/Darth Gurden
The Voice of Reason and Sith Lord
“Putting the FUNK! back into Dysfunctional!”
Coming soon. The USS Ganymede NCC-80107
"Ad astrae per scientia" (To the stars through knowledge)