I think you're right, they'd no longer be an FA club so no more FA Cup.
I also think you're right about the English sides probably staying out of such a league. The Premier League has better parity than most leagues in the world as it is, very good attendance for the top clubs, and also get lots of TV viewers around the world.
If Chelsea or any of the other really big clubs joined a Euro Super League... What would they gain? Maybe more TV money, but also I'd argue lower attendance (less away fans) and much, much higher travel costs. Is it really worth it?
For English teams, the current European format is better.