I totally thought that the whole Lin thing and then Tenzin's family getting captured was a total Emperor's New Groove moment when Kronk and Yezma somehow passed Kusco and Pacha. XD it felt like that just happened out of left field for me or something.
Was it worth it? Hmm... Well, she had the right idea and right mindset. She wanted to protect Tenzin and his family because she loves them all (clearly deeply). Definite character development there.
I suppose it would have been worth it because honestly, she provided a distraction for Tenzin and his family. She preoccupied Amon so they could get away, even though it cost her her bending. And then when she said she'd never reveal where Tenzin and them went, she demonstrated her love and devotion to them. Which, more character development.
But yeah, other than that, maybe it wasn't really worth it. I think it was added in to demonstrate Korra's abilities and give some character development/depth to Lin.