The faith have gotten more power than ever by having King Tommen on their side. If the faith declare him a product of incest they can’t be sure that the next monarch would be as easy to control as Tommen.

Didn’t a number of the BWB start out as Stark banner men who were sent to the Riverlands to catch the mountain before Cersei moved against Ned? Is always thought that was why Beric gave up his chance of resurrection in favour of his Lady.