^Because that's just one facet of her job, in addition to being a check on the captain's judgment and an advisor in diplomatic and contact situations. Understanding the psychology of an alien race or a hostile captain can be essential to resolving a situation peacefully and safely, so I can certainly see the value of having a psychologist as a regular advisor. The show just didn't do a very good job of depicting how that would work beyond having her say "I sense anger from the guy who's shouting and pointing big guns at us." But we did see her utilized more usefully in some contacts, like advising Picard on how to address the Jarada and avoid a diplomatic screwup in "The Big Goodbye," or explaining what was known about Romulan culture and psychology in "The Neutral Zone."
I don't know if the idea of a sort of "captain's advisor" makes a lot of sense. First, if your Starfleet captains are supposed to be the best of the best, it doesn't show a lot of confidence in that captain to need a counselor right there for hand-holding. Secondly, the XO does much of that sort of job already, advising the captain on the situation, providing advice, etc.
Finally, captains are supposed to already act as "ambassadors" of a sort, since they do so many first-contact missions. Picard was an accomplished diplomat. I think it just makes more sense if Troi was on the bridge because of her empathic abilities for viewscreen encounters, and NOT because she was ship's counselor.