What are the pros and cons of using different transistor types in a Colpitts oscillator circuit?
Is there any reason MOSFETs in Colpitts oscillators seem so rare?

The most basic problem with a MOSFET in a colpitts oscillator is that the gate-source junction is much more capacitive than either a JFET or a BJT (like a hundred to a thousand times more). So, that gate-source capacitance throws a big wrench in the works.
I mean, they can work (sub 1 MHz for instance) but due to the unknown value of the gate-source capacitance, the frequency they produce is a little unpredictable and, with varying gate bias conditions, the capacitance can also vary and add further woe to the performance and stability. After all, you want an oscillator to be stable and predictable.