Yes, brown wire is positive power for pump.
If your breaker is not tripped and if the redundant fuse aft off the battery is not blown, then something is open, which would also explain no light at switch, incomplete circuit.
Looking at my diagram, each cranking battery should have a fuse for a bilge pump. 1 battery for one pump and the other battery for the other pump. But could also be on the house battery. Wires get moved sometimes over the years. I hate those inline fuses stuffed into nowhere land, all fuses should go to a fuse block.
If fuse is blown, sometimes the filament just fails (very infrequent) and you change the fuse and all is well. But if it blows a second time right away, you have a short, which is power shorted to ground somewhere.
Usually that would not be the float switch as it just opens and closes positive power to pump, no ground wire associated with float switch.
Common for bad float switch is an open. This is not likely your case because your pump does not work from the dashboard switch either. When your float switch is shot and open or broken wire, the pump should still work from the dashboard switch.
If float switch stuck closed, which also happens, debris can get stuck under it or it just gets too caked up from an unclean bilge, then the pump would run continuously, can kill a battery, can overheat the pump and blow a fuse that way.
There are more than a couple things that could be wrong.
So, we need you to locate the aft fuse and take it from there.