Just one step in troubleshooting would be to check and verify that the generator is putting out 120vac. I have a 5.5K so I do have an outlet, if you don't, It is simple to remove the cover from the ATS and hook a meter to the cables from the generator and see if the generator is outputting and eliminating geting into the guts of the thing.
Doing that would eliminate one big possible problem.
If the generator is not generating 120 vac, the generator engine will not run, so check that the delay circuit is receiving power. As DrewE said there will not be any power to the system until the delay circuit closes and pulls in the ATS but the generator produces power as soon as the engine starts. Actually it starts producing power as soon as it starts turning. That is one of the first things the tech in the authorized shop did when mine failed.
What you have done is to make sure that the generator is outputting. Like I said, mine was not and threw a code 32, Low Oil.