Light should come on regardless of whether the flame lights or not—there's a different pattern for working and not working.
After the idiot's delight fix doesn't work, the fuse is the next simple thing.
Do you have a wiring diagram with the stuff you got when you bought the trailer? If so, that can help with troubleshooting.
If you don't hear the flame and the light doesn't come on, it could be a fuse or a loose wire on that part of the circuit that operates the light and the igniter. If you don't have spare fuses, borrow one you know is good for the water heater fuse. I believe the 12 v. circuits are identified on the inside of the converter cover under the bed.
Then it is loose wires. And cleaning the igniter—though I'd think the light would work regardless of the condition of the igniter.
I think there's a circuit board to control the electrical part. That can be shorted out by bugs, dirt, dust, or cracked, or otherwise blown.
Since our trailer is the same age and trim line as yours, and I have seen people reporting problems with their water heaters at about 3 or 4 years, I guess I should be looking ours over and clean parts and check them.