Regarding Betazoids... maybe most don't like the idea of serving on ships full of people without telepathic abilities. Or a taboo, perhaps.
Also possible, regarding Troi being picked... maybe Starfleet, or Picard, thinks it better to not have a full telepath as Counselor and part of diplomatic affairs. I can see an alien race in a negotiation being offended having someone who can read minds as part of negotiations. Wars have started because of less. But an empath like Troi can be that middle ground that could be advantageous enough and not offending enough to be grest at the job.
Also possible, regarding Troi being picked... maybe Starfleet, or Picard, thinks it better to not have a full telepath as Counselor and part of diplomatic affairs. I can see an alien race in a negotiation being offended having someone who can read minds as part of negotiations. Wars have started because of less. But an empath like Troi can be that middle ground that could be advantageous enough and not offending enough to be grest at the job.