Jamie didn't do it because of Frank; in fact, his explanation for not telling Frank about taking the exam was because he didn't want Frank to feel like it was because of him.

It wasn't becoming a sergeant that was the problem, the issue was how the writers approached his stories; over time the writers gave Eddie more and more of what had been Jamie's viewpoint. Their reasons for moving him to Intelligence remains the biggest mystery as most of the writers seemed to have no ideas for that role.

I don't think they were a replacement for Danny and Linda. Their relationship was presented very differently than how we saw Danny and Linda's marriage in the early seasons. The biggest issue was that Jamie and Eddie's relationship was usually shown through work, we saw more of them together at the precinct than as a couple.

This article misrepresents what happened with Jamie and Eddie's working relationship; Jamie changed precincts when he was promoted to sergeant, and Eddie followed him. Plus, when McNichols was (oddly) promoted to captain, Jamie was transferred to a different department. McNichols was the same rank as Jamie in seasons 10 and 12.