Winnebago used Atwood, I know how that one works. What you are seeing at the switch suggests you have power to the controller board, connections to the switch, and at least part of the controller board is working.
I'll let you borrow my basic diagnostic skills.
Possibilities at the water heater:
Gas valve not opening. You'll hear the clunk when it opens. No clunk, possibly a faulty valve, or controller or connections.
Gas valve opens, but no spark. You can sometimes even hear the spark. Can be the sparker/flame sensor, connections, controller. If it is trying to spark, probably connections between controller board or sparker, or within the sparker.
You get ignition, but it shuts off the gas after a few seconds. That is a flame sensor problem, in the controller, or connections to sparker/flame sensor. I had this problem when the high tension wire to the sparker had a discontinuity. The spark jumped the gap OK, but there was no continuity for the tiny low voltage current used for flame sensing. Needed to replace the sparker, a cheap mechanical part.
Then the more basic problem:
Maybe no gas supply to water heater, shut off at tank or air in the lines.
Diagnostic skills at shops are sometimes problematic. Often they'll replace, or have you replace, the controller board, even though it seems to be functioning for other tasks. Apparently it is easy to replace and has a nice margin for the parts department. If that doesn't fix things, they'll start looking at things like maybe the sparker is defective, or the gas is not turned on. When they find something else to be the problem, they'll leave the new controller in, after all, they just sold it to you.
Atwood DSI heaters do not use a temperature sensing device to detect a flame. After sparking for ignition, the controller applies a voltage to the sparker. If there is a flame, the ionized gases will allow a very small current to pass. Any excessive resistance, like a bad connection to the sparker, or a break in the high tension wire, will prevent the controller sensing this small current, and it will shut the gas off, then retry lighting a few times, then shut off for good.
Atwoods do use a temperature sensor to limit how hot the water is heated. That's not in the flame path, it is in the side of the tank. That will usually not prevent the heater from lighting, but may prevent it from heating to a high enough temperature, if defective.
