"Next weekend" I didn't do what I wanted to. Of course that other project couldn't be finished in time ... But it is mostly completed now, so I was able to clean up my bench today and get started.
I do owe you the report (though probably no one's reading this thread anymore), so here it is.
1. Tek (RCA) transistor: as before, about 70 mA DC collector current.
2. Small cap between C and B: no change at all.
3. Siemens 2N3055: 44.3 kHz oscillation, needs more base bias for correct amplitude, about 80 mA collector current.
4. Darlington (BC107B, too much gain): hefty oscillation already at small bias, transformer makes high pitched noise, collector current around 1 Amp! I aborted this after a few seconds.
5. Back to original Tek transistor: stable 44.3 kHz oscillation, about 80 mA collector current. Transformer is slightly audible.
All these tests were done with a breadboard (hardly needed for the 5 odd components) and no transformer loading at all.
The whole setup was not mechanically sensitive, so I'm quite sure I had no bad connection. Knocking the transformer also had no influence.
I've seen my share of strange things in during the years, but what is this supposed to be? A part time transistor?
Is it time for throwing the towel?