I don't like using non-core base classes. I think there is no character concept that really can't be done outside the core set of classes, so the reason to use a non-core class for me would be for an alternate mechanics, but so far there hasn't been anything among the published non-core classes that stroke me as interesting to add to a game (with the exception of the old Mongoose's Chaos Mage).
The Psion could definitely be an interesting class, but I've never been interested enough in the whole psionics subsystem to buy an entire book and study how it works. It has in fact a "different concept" but I don't know why I always feel it slightly inappropriate for a fantasy setting (I'm probably too traditional...). It's be great in a modern or scifi setting tho.
The Healer sounded like the ultimate class which is there to make everyone's favored NPC or cohort, but it probably wouldn't appeal many as a PC.
The Warlock and the Warmage for me are narrowed sorcerers and feel just plain boring. No new concept at all IMHO.
The Spellthief is too specific for my tastes as a base class, although its signature abilities would be nice if available as feats or as a prestige class.
The Scout seems nothing more than a variant ranger. No new concept at all again.
The Hexblade also has no new concept at all, being just another take to the fighter/mage idea. The Swashbuckler is more of a variant fighter than a new class as well. However I think these two base classes probably work well mechanically. It's just that instead of having new classes (of which I'd have to take every detail), I'd rather try to make these concepts with the core fighter and some added feats or special abilities.
Anwyay, I would allow a non-core class if a player really wants it, but I definitely don't suggest its use by e.g. giving players a list of 'allowed' non-core classes. Instead I largely prefer to make variants of the core classes. E.g. instead of introducing the Favored Soul, I would allow UA's spontaneous cleric. I think there are several advantages of using variant core classes instead of non-core classes, for the fact that such a cleric is still a cleric for any purposes (for example, using spells and stuff from supplements).