This is my new head cannon for Lin
I think at some point in time, Lin was very free spirited and independent and not at all rigid like she is today. She had a close friendship with Aang, and she seems to have been very close to her mother. She did learn all of Toph's cool bending, like sensory bending and metal bending; Toph doesn't strike me as the kind of person who would teach someone who a) didn't have the inclination or b) wasn't very close to her.
So I see a teenage Lin as being maybe a bit Sokka-like [not saying I'm into Tokka, I'm just saying personality wise]. Pretty free-spirited, flexible and adaptable. Not at all rigid or a follow or set the rules kind of person. She and Tenzin matched up pretty well that way. But Tenzin, for an airbender, follows rules and traditions pretty closely. Being with a wild child may be cool at first, but eventually that gets annoying, and seeing as he's the only airbending child of Aang, he's got to settle down. Pema seems very relaxed about things, and I wonder a bit of bending may have been at play... by being with a nonbender, Tenzin might have hoped that his children would be airbenders or nonbenders. With Lin, they might have earthbenders. Not that Tenzin has anything against earthbenders, but the third path would have be taken care of.
So Lin, after being dumped by Tenzin, decides to 180 her personality. She decided to ground herself as much as possible to ignore Tenzin blowing and whirling about her. I'm the kind of person who, when I'm upset, I throw myself into a project. Maybe Lin is too. So the creation of the metalbending cops was Lin's project. I wonder if Toph supported it, or if she had passed and part of Lin's motivation would be to see that her mother's bending techniques didn't end with her. I read Lin's protectiveness her cops as not just a leader, but a teacher as well.
So yea. Tell me if I'm completely off hte mark hahaha.