By the way, the whole conversation Azriel had with Rhys in the bonus chapter of ACOSF makes me think that, should Elain choose him, if Az's mate ever showed up, he’d leave her immediately. I think he actually just wants a mate. It wasn’t about Elain at all.
Also, that's probably why Rhys was so weirded out by it. Feeling entitled to a mating bond isn’t right in the first place. Rhys never felt like he actually deserved it. And Azriel's approach to the Elucien mating bond is similar to how Tamlin acted toward the Feysand bond.
Rhys put it on politics probably because that was the most logical angle to consider, but the truth is, he’s known Azriel his entire life. He just doesn’t want him to make a mess, especially when it’s clear from what he says that he’d leave Elain the moment his mate shows up. Rhys also seems to have a soft spot for Elain, so maybe he just doesn't want her to get hurt. There's also the possibility of Elain choosing Azriel and then breaking his heart by changing her mind about wanting to explore the bond with Lucien. Azriel is one of his brothers: he may not wish for his heart to be broken as well. Additionaly, Rhys has a bond of his own, and maybe he doesn’t feel right denying Elain the chance to eventually get to know Lucien and experience that bond herself, should she want to. He had also accepted the idea of Feyre marrying Tamlin, so he knows what it feels like to be rejected by a mate, and he might sympathize with Lucien on that side of things.
Moreover, Rhys has stated multiple times that the way people treat Elain is unfair, as they are constantly shielding her, thus preventing her from truly figuring out who she truly is, especially now that she is fae. Elain has made it clear that she wants to prove herself and has things she needs to deal with and figure out. No one is listening. Nesta doesn’t accept that Elain no longer fits the image she has of her. Feyre still thinks Elain needs to be taken care of. Azriel thinks he knows how she feels about Lucien and their bond, and what she wants. I think Rhys is one of the few who truly understands that. He was the one who helped Feyre open her eyes to the matter. His entire approach to anything related to Elain focuses on enabling her to make her own journey on her own terms: discovering who she is, choosing her path, and making a conscious decision about the mating bond rather than a rushed one. Which, by the way, is what I think her arc will be about.