On a ship, with unsuitable for duty people being screened out, the mental issues you would have to deal with are either things which are undetectable early on, or products of their time on the ship/things which happened afterwards.
So Angela Martine, after returning from bereavement leave, might need ongoing counseling for her fiance's death, which of course was not a factor prior to her Starfleet career, the way a person who has OCD or some such issue would be, having had it for some time.
Similar to the difference between a genetic condition and an accident. Both might leave someone paralyzed, and both would require therefore a neurologist and anyone else associated with such a condition, but for different reasons.
On a Starfleet ship, you'd have people who specialize in acquired conditions, if there is any measurable discrete difference between that and genetic ones in 22nd-24th century medicine.