Don't forget, Kara was all about finding validation from Clark early on. He gave her advice and support way before he appeared on the show. Superman was THE man when Kara was still finding her legs. So yeah, in this universe they're a little more even than you'd like - they support one another and Superman can have a human moment and lean on his cousin sometimes. To me, Clark's moment spoke to his sense of super-altruism. He thought he could focus on himself for once and things went to shit, so he was racked with guilt in a way that nobody else would.
And also yeah, Kara outlasted him in the "recharging the solar panels" scene, but he'd been on Argo for a long time under a red sun.