The thing, for me, about the differences in reaction between Elain being kidnapped by the Cauldron/Hybern and Gwyn getting stolen off to the Blood Rite is that Azriel's reaction doesn't mean he doesn't care about Gwyn. He pretty clearly DOES care about all three of them, particularly Nesta (we've seen their actual friendship ON PAGE, after all, as recently as HOFAS). Azriel choosing to NOT go after the Valkyries isn't because he's like "🤷 whatever, they'll be fine," it's not even "they're strong enough to handle this," it's Azriel doing something the antis like to claim he's doing elsewhere - - - his job. His job, for HUNDREDS OF YEARS, before Rhys was even High Lord, is to protect the Night Court. He is CHOOSING to do that job over his own emotional wants (help the women he doesn't want in that situation).
When it comes to the Elain rescue, we're supposed to see that as unusual. We're supposed to see it as unusual at the time, and we're supposed to see it as unusual in contrast as time goes on (in my opinion). Azriel CHOOSES to save Elain over doing his job in that moment. It isn't even a question for him. He would have gone without Feyre, he would have gone despite being told not to, he was going to go get Elain back. Even if it meant his death. It doesn't make sense, really, unless there's something else going on there. Unless he has deeper feelings for Elain than even just caring about her. Unless he, maybe, is Elain's mate (either because she has more than one or because he's her "real" mate).