Still - I think the analogy is suiting. Martin Luther started a movement which granted common people access to religious texts (by translating them from Latin). Sci-Hub removes other obstacles to access knowledge, but the goal is somewhat similar.
This is not to say that peer review and expert validation doesn't have it's place in the process - but it could (should) be done in open, with as many eyes watching as possible. So IMHO anything that denies general public access to scientific work should be removed.
I don't think the analogy works. Luthor took a single text and translated it, making it accessible. The fact that it was accessible didn't affect the creation of the text.
By contrast "the interpretation and approval of academics from high-status institutions." is an intrinsic part of the creation of the document. It's part of the peer-review essential to the creation of good science.
Experts disagreeing on the fundamental nature of the holy trinity is not the same as experts disagreeing on the experimental design behind testing a new category of drugs.