This is the long-term strategy I would be running if I were on the side of those christian lobby groups that have been successful recently.

I think you’re right and this is the strategy they’re running. They use the overtly bad stuff (pedo loli shit, for example) as a Trojan Horse (or maybe a wedge / pry bar) to slowly smuggle in functional censorship of LGBT content (plus anything else they don’t like). These don’t have anything in common, to be clear, other than being “taboo” according to some.