The bridge officer's test is probably exactly what it sounds like - a test to determine an officer's readiness to serve on the bridge (where they could at any time be called upon to take command) or to stand a watch (where they would definitely be left in command). It's just that Troi, as a staff officer, hadn't taken the test, while even the greenest ensign wearing gold or red had taken the test before leaving the Academy.
The facts that Troi took the test, passed it (eventually) and then was promoted may not be as directly related as they look. At best we can say they're correlated. Was Troi perhaps up for promotion anyway? Would she have been promoted to commander if she had done something else to earn points? We don't know. We do know that Pulaski had the rank of commander and was definitely not a bridge officer.
Surely capping blueshirts at lieutenant commander is unfair and unnecessary; there are plenty of reasons to promote a person even to flag ranks without them having starship command experience. Commodore Stocker made it to flag rank without having commanded a starship, for instance.