Why are BJTs common in output stages of power amplifiers?

To make an Audio Voltage source, you want the crossover voltage distortion to be null which requires some quiescent DC current > 1% of the max current. This modest distortion and output impedance is reduced further by negative feedback or excess open loop gain. The active diode bias DC voltage can be predicted in mV for the differential Darlington output stage.

However for MOSFETs the conduction threshold can vary 50% e.g. 1 to 2V or 2 to 4V, so the biasing for cross conduction to eliminate crossover distortion is not easily done with low voltage gain linear power amps.

Edit May 22:
Also, Thermal Runway exists as stated by @Thor from the micro-array FET structures sharing-current with Vgs NTC effects in linear mode yet PTC effects for RdsOn in full conduction mode. Without proper transistor component selection, this can result in catastrophic failure.

MOSFETS used to be more common in power amplifiers, but they were often the lateral type power MOSFETS.

Most modern MOSFETs (Vertical MOSFETs / HEXFETs) are highly optimized for switching and require very careful design in a linear amp design. For example these modern switching types have a large nonlinear gate capacitance that is difficult to drive.

In addition the likes of HEXFETs can suffer from localized heating effects that can cause thermal runaway in a linear application.

A good description of these issues can be found here

Lateral MOSFETs are still available but are more quite expensive. See here

So really it's not a case that MOSFETs cannot be used, but it's often more difficult and less cost effective to achieve the same performance and reliability for a given price point.

Second Breakdown

(Many) Audio amplifiers operate the output stage in their linear region.

Modern power MOSFETs are not designed to operate in the linear region. Many of them (HEXFETS) are composed of a grid of hundreds of thousands of smaller FET elements to increase power density and switching speed. Other switching-optimized MOSFET families have similar constructions, with large die areas and/or arrays of smaller elements.

For MOSFETs, the threshold voltage has a negative temperature coefficient. As a particular area of the die / FET element gets hotter, it's threshold voltage decreases and since the MOSFET is operating in it's linear region, that area conducts a larger portion of the current, so it gets even hotter. Before long, the localized heating on a tiny fraction of the die has resulted in a short circuit, often called "Second Breakdown".

But...

A relatively new type of amplifier, the "Class D" amplifier, works by switching the output stage transistors on and off rapidly, at a frequency much higher than the speaker is expected to reproduce. A low-pass filter filters out the high-frequency noise, and amplification is achieved through varying the duty cycle.

MOSFETs are extremely common in such designs, as class D amplifiers either have the output stage elements fully on or fully off. As power MOSFETs are optimized for that, that's what they are used for.