The benefit for buying psi specialties over buying regular dots seems to be that by the time you're buying specialties, there are no more baseline dots.
Well that and 3 weeks is faster than the 4 needed to get to rank 4 in any psionic skill.... though the specialization would need to be really, really useful to justify having it over a rank 4 base skill, since just using Teleportation as an example, we're talking about the difference between merely teleporting vs warping space and making permanent portals.
Can Astral Projection be used for non-Psionic training to deal with our training time problem (say, Lore) after we get 1 dot?
Training is apparently a matter of concentration, which is why full training time overlap isn't a thing. Since I'd assume the Astral Projection itself requires concentration, we'd be splitting attention between it and whatever else we were trying to train while using it.
What it WOULD help with is with eliminating travel time and potentially needing permission from adults to go places, if we could just sneak a Projection to the alternate location while we were supposed to be, say, in school. Making it possible to train Medicine at JP's while at school - but Amu still wouldn't be paying attention to class if she was focused on the Projection at JP's, so if we were trying to study Lore at the same time there, we'd just be dragging out the training of both activities.
I'm also pretty sure it wouldn't help with Integrity, which is possibly the most critical stat to train. We've been told Amu can't do more than 2 hours of it safely per day, no matter what.
That is to say, we could probably get some good training overlap going between Integrity and Astral Projection, if we, say, conjured a Projection and got Ran or Su to attach themselves into it (like how Akkun is currently in the driver's seat of Ami's projection) - but if we wanted to keep the Projection going past 2 hours, they'd probably be forced to "detach" themselves at that point and then our Integrity training for the day would be over.
(....Which actually makes me wonder if keeping Ran or Su out itself qualifies as training Astral Projection. Huh.)