You're welcome.
Its not my logic, it is however code. Look up NEC 702.4 and make sure you understand the rules for adding a generator utilizing an ATS.
And yes, you are correct that you would need a large generator for handling a 200A service.
I did make make the assumption that you were doing a whole house setup based on the size of the ATS. Just want to offer the info that you can't do that and meet code with a 10kw generator.