Wow, lots to digest.
First, as a 50 amp MH you normally have 240 VAC to the coach. Probably nothing in the coach uses 240 VAC but your breaker panel distributes the power as two legs of 120 VAC (the legs are 180 degrees out of phase) just like most things in your house. In fact, your generator likely just supplies two legs of 120 VAC (both legs in phase) to the coach as well.
A 30 amp RV receptacle is a single phase of 120 VAC just like the 15/20 amp receptacles in your house, but capable of supplying more current.
A 30 to 50 amp adapter (dogbone) will take the single 120 VAC hot leg in a 30 amp receptacle and supply it to both hot legs in the MH connection. By connecting the adapter to an incorrectly wired 30 amp receptacle with 240 VAC to it you probably essentially shorted out the two hot legs or at least one to ground and should have instantly tripped the two supply breakers. The adapter should not have passed damaging electricity to your MH, but who knows? As we have no idea had the incorrectly wired receptacle was really wired. It could have passed a 120 VAC hot leg to either the ground or the neutral leg in the MH.
But on to troubleshooting/repair:
First - replace the 30 amp to 50 amp dogbone if not already done.
Second - you replaced the ATS, but SP still doesn't work (but generator power does.
Realize that the ATS is generator dominant and SP defaulted. That means that with no power applied the relay allows connectivity from the SP inputs to the output on all contacts (this is also true if only SP is applied). When the generator starts and supplied voltage to the ATS there is a delay (30 seconds to a minute) and then the ATS will switch to generator supplied power even if SP is connected. Make sure the new ATS is connected properly and with no power applied make sure you have connectivity on each of the contacts from the SP input side to the output side.
Next, connect your SP and measure voltages at the ATS. You should have 120 VAC on each of the hot legs (L1 and L2) to ground and to neutral. If connected to a 30 amp receptacle you will 0 VAC from L1 to L2. If plugged into a 50 amp receptacle you will have 240 VAC from L1 to L2. You should have the same voltages on both the SP input side and the ATS output side.
I suspect you will find your problem by this point. The only other issue may be your surge suppressor/EMS. Is it hard wired? Is it on the SP side or the ATS output side?
I see you added the ATS info to the OP:
The Furrion 50 AMP ATS is NO CONNECTION DEFAULT which means with no power applied there is no through connection. When SP is applied the relay latches to the SP position. So it is possible you don't have 120 VAC on both L1 and L2 due to either a wiring issue or a bad dogbone.
Ted & Melinda
2016 ACE 27.1
2016 Chevy Sonic Toad
2020 Chevy Colorado Z71 Trail Runner Alternate Toad