score:8
So it turned out to be fairly simple. The ESTA application and the marriage certificate were enough. Our marriage certificate was checked when we checked in at the airport. They didn't ask for it at customs in the US (I suppose the custom agent took our word for it even though we don't share a last name). It all seemed fairly routine for United Airlines. It's definitely not necessary to get a B1/2 visa. Since I traveled together with my spouse, I am not quite sure how one proofs being married to a US citizen when travelling alone.
Generally, I would recommend to contact the airline to find out what they need to let you travel.