The short answer is "no". While she never proved as useful as she could have been, Troi was there to provide empathic information to Picard. In terms of her job description, a "therapist" fulfills Roddenberry's edict that mental health will be as important as physical health in the future, especially as we understand the mind better, and I agree with that. So its no on both counts...
RAMA
However, does it make sense for her to be on the bridge the vast majority of the time? A therapist and a diplomatic aide are very important yes and depending on the mission, probably warrants a spot on the senior roster, but they're called for specific purposes that don't happen very often compared to other bridge positions. And frankly, the times when the Enterprise comes across another ship or anomalies with mysterious new life forms are one thing, but that's a small minority -- only slightly more frequently are coming across phenomena with no life forms that can be studied with a simple probe or buoy, but the vast majority of the time is spent traveling, delivering supplies, meeting convoys, etc... things that are usually too mundane for an episodic plot and things that happen between episodes. Days or weeks can go by between episodes, sometimes between acts within an episode. If the Enterprise is at warp for 6 days and encounters an enemy on day 7, it makes sense for Troi to be there on day 7, but the show implies that she's on the bridge when nothing's going on, aka the other 6 days of the week.
I can accept that Roddenberry ruled that mental health be just as important as physical health because that's a very real concern today, but if that's true, why does Deanna get a bridge post but not Crusher? If the answer is because Crusher has to deal with the well-being of 1,014 people, then the same is true for Troi as well (after all, the show established that she makes appointments, sets aside time for crew, and devises treatment -- just as a real therapist should). That's on top of research and studying she'd have to do for anticipated diplomatic missions as well.
The captain might be hogging Troi, when really the rest of the crew and future missions might be depending on her! Now *that's* dating it.
Also, let's clarify something: Picard keeping Troi on the bridge because of her empathic abilities is different than Picard keeping a counselor as a bridge officer. There are non-empathic/telepathic counselors in Starfleet -- do they get bridge positions because they're counselors? Are they denied a post because they're not psychic? Keeping someone around for their job and keeping someone around because of their unique powers are two different things, and could be something of a cheat. I get a sense that people in this thread are confusing the two: if Troi were a mere human, would she be as crucial for many of Enterprise's crises? Would Picard go to her as often?
The Enterprise-E probably would've been crippled much sooner by the Scimitar if Troi didn't lend a hand -- but that was b/c of her telepathy, not because of her counseling skills. An equally skilled but non-psychic counselor would have been useless at the crucial moment. Obviously having a psychic on the bridge at all times makes sense, but not a counselor.
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