Increasing the supply voltage and the emitter resistance can help with this. Also making sure that you don't have any mid-length time constants around for squegging to find is good idea.
They don't tend to have the problem you descibed above but they are not without their own troubles. With JFETs, you need to be careful about the bias to make sure that you don't have it driving the gate into conduction. A JFET can have a very high gm near Idss that still increases as the gate goes towards forward bias. "Gate leak" bias tends to yeld a noisy oscillator.
If you control the average emitter current, you will likely do even better. For large signals, you get a reduction in gm as amplitude increases. Basically, any gain that goes to making harmonics, is making as much fundamental.
[...]I've seen that happen. It can destroy the transistor if nothing stops the run away condition.
[....]You need to solve the positive feedback issue mostly before you try to close the loop. You can make your RF amplitude detector circuit as a combination of a fairly quick inaccurate one and a slower more accurate one. You can stand a 10% variation in the "P" term of a "PI" controller. In the long term it is the "I" term that sets the amplitude.
[....]What is the swing on the base? I had assumed it would be small since you are doing an AGC. Decreasing average base voltage with increasing signal is a sign of nonlinear operation.