“Purport”. That’s a good representative word and I’m glad you used it.
And that is one reason I reject Lumen Gentium. Lumen doesn’t use the word, “purport”. Lumen states a supposed “fact”, that in fact is false.
You reject Lumen Gentium, a magisterial document adopted by an Ecumenical Council and approved by a Pope? I’m sorry, a Catholic cannot simply “reject” teachings of the Magisterium like that. You are apparently a Cafeteria Catholic at best.