It just does not work to safely and nicely connect a 120 generator to both sides of a 240 house and make that work safely. 240v is a single phase, the 120 we use for smaller items is split off of that; it is not a phase of its own. So you are trying to join 2 things that already are just one thing, it gets real messy real soon.
If this were your plumbing, all your water supply comes in on one pipe. Some goes through the water heater, and some just goes on a parrallel path. Same water, but down the line one side is hot, one side is cold, and you can no longer interchange them or you have a mess.....
Your question on powering one side of the house, one of the 120v sides with a transfer switch is an interesting one. I don't understand why that wouldn't work, but I never really gave it much thought. Something to think on while on the tractor again.... Off the top of my head, wouldn't any 220v items flow the 120 across to the other side, creating problems? Hum.