Yes, Finrod's actions here are...interesting. Surely if anyone were to return with Finarfin it would be Finrod? He was somewhere around Turgon levels of reluctance to go, but still left.
At some point, we are likely going to show a flashback to his farewell with Amarië. That could be here. In the flashback, he could give her his reasons for wanting to go to Middle Earth. If we don't do it here, it's going to look like it came out of nowhere when we do it at the Feast of Reuniting in Season 4.
Finrod is also (likely) the wisest of the Noldor who go to Middle Earth. He's...a philosopher, of sorts, eager to learn new things and meet new people, but able to synthesize all of that into his world view. He has some sense of prophecy (not unlike many other elves), but also can pull off the magic-by-force-of-will-and-song trick. Sure, Sauron defeats him, but Sauron is a powerful Maia, and Finrod at least gave him a run for it and didn't cave instantly. It is very difficult to picture Finrod getting mad at anyone, ever - he's a very calm and even-keeled guy who can interact with people who disagree with him without taking it personally. He befriends Men and Dwarves and the Sons of Fëanor and Ulmo - ie, he gets along with *everyone*.
So he has to recognize the Doom of Mandos for what it is, and to recognize that their entire quest to defeat Morgoth in Middle Earth is doomed. So why does he keep going? And how much sense of dread/foreboding/doom/fate does he have about the entire enterprise? Could he be using Ulmo as a model - Ulmo often acts independently of the other Valar, but that does not mean his actions are wrong. So maybe he thinks there's a way to rebel honorably without being guilty of all the things Mandos warned of? I do think we have to hash out what Finrod is thinking, and it has to be nuanced and involved, not just a single reason. He very much could be swayed by someone else - his sister Galadriel or his cousins whom he is close to. But at the end of the day, he's thought this through thoroughly and is not just blithely going along without due consideration.
[As for the brother/son inheritance question, do keep in mind that Fingolfin is going to be recognized as Fëanor's heir, Turgon is going to be Fingon's heir, and Orodreth (who may be a younger brother or possibly a nephew) is going to be Finrod's heir. So, I don't think we have to assume that it's the son rather than the brother who automatically inherits in Noldor culture. I realize that male primogeniture is common enough as an inheritance pattern, but it's not like we don't have plenty of counter examples, even down to Castro naming his brother as his heir in Cuba in this century!
We aren't going to discuss Maedhros' ceding the crown to Fingolfin until next season, but it is quite likely that he pulls a sleight of hand to do that legitimately. He calls Fingolfin 'the oldest here present of the House of Finwë',' potentially recognizing Fingolfin as *Finwë's* heir, thus ignoring Fëanor's claim to the throne of the Noldor (which was never very official, now was it?) So in that sense, it's still a son, not a (half-)brother who inherits, and explains how the entire House of Fëanor is disinherited, even to the point that Celebrimbor is going to have to acknowledge Gil-galad as High King! But it also calls into question how significant age is as seniority in whatever inheritance plan we set up - after all, elves don't have to worry about a younger brother dying of old age, and so a brother or a son or a nephew would all be equally likely to be healthy adults with full faculties. But again, a discussion for another time!]