Jake.
I'm no expert but I think you are correct to clean your air guns with felt pellets.
Airguns don't burn gunpowder and inherit the associated grime.
They don't shoot copper jacketed projectiles, so the rifle lands do not get the associated coining.
I really see no need to run any type brush through the bore, or even a patch for that matter. You could do more damage to the rifling with a cleaning rod than any benefit gained by the practice.
Felt pellets work very well. They start out dirty but come out spotless after just a few shots. When they come out clean, your bore is also clean.
I did push about 20 felt pellets with Flitz polishing compound through my Supersport when it was brand new. The barrel was filthy. It appeared to have some sort of rust preventing stuff in it. Anyway, I was so anal about the bore that I used a thin wooden dowell as the rod. When I finished, the bore was mirror brite and remains that way even today.
When I put my Sheridan away after shooting, I put a couple of pumps in the chamber. That's all that is necessary because the barrel is brass.
Before I put the Supersport away, I fire one oil soaked felt pellet through it. It's barrel is steel and could rust without protection.
Bill