I've been wondering this too. I just booked on chase travel for first time to get 10x points and I saw prices on there that were cheaper than on Marriott. No Marriott. But it's made me think about whether the sites are cheaper and I can't figure out how to do a BRG.

I'll keep booking via Hyatt because the benefits and fixes seem good (and I don't have lifetime) but will book on chase sometimes intead of Marriott of cheaper if the benefits aren't great anyway.