Doing this has been common practice at every flying school I've worked at, I also did it during my PPL training, so maybe I'm just pre-conditioned into thinking it's normal.
Other than my current establishment, it was done just when the FI thought the student was up to it, sometimes IIRC pre-solo/medical. At one of them, we had a share of the MoD FSS contract for a few years and I don't ever recall CFS disapproving of the practice.
Nowadays, I'm less sure it's a good idea. We do allow it, but have a "pre-solo engine start-up" check sheet in the individual student records. If we're honest I think it's mostly for a**e-covering purposes in the event of an incident. It contains such items as, familiarity & competence with normal & emergency procedures and being legally entitled to act as PIC (medical & age).
I do think it has some benefits in building student confidence and self-reliance, but only up to a point. Also, if we're being honest, I've done it as a time saver, although that's really in the past, don't let myself succumb to time pressure these days. Having said that, is it an unnecessary risk? I may have to reconsider whether to continue doing it.
I am aware of how badly wrong it can go, the place I did my PPL had a couple of wrecked aircraft sometime before I started, when a student started up with loads of throttle and no brakes and drove their's into another one across the apron. Surprisingly perhaps, they didn't stop people doing it though, I don't think there was any problem with the insurance or trouble for the instructor from the CAA. That was decades ago though and we live in a very different regulatory and litigious environment now.
Very interesting point regarding getting in/out with prop's running, it seems to be common practice, especially when conducting dual to solo circuit training. As has been said, we shouldn't allow students to think this is acceptable practice, other than in certain controlled circumstances. As we've seen of late, this can have horrendous consequences.
However, don't we have to accept we're going to do one or the other of the above "bad practices". If we don't shut down, then we are forced into an engine running crew change. If we close down to get out, then we're going to need to let the student do a solo hot start. We can't avoid both situations, can we?