OK, I've now played around with replacing the pass transistor with a MOSFET (IRFZ14, pretty standard, I think) and using a TL431 as reference instead.
The results are sobering, to say the least.
The TL431 gives no problems compared to the the TLV431, so that's fine.
But the combination TL431+IRFZ14 has produced one of the most powerful oscillators I've seen. And it's impossible to tame.
I was sceptical to start with, but that was only my "gut feeling" Now I've seen it in real life.
This is the circuit (ignore the gate resistor, it makes no difference). Feel free to try it out yourself: