I just looked up the BBG Wireless schematic:
https://github.com/Seeed-Studio/beaglebone-getting-started/blob/BBGW/Docs/Hardware/BeagleBone_Green_Wireless_V1.0_SCH%26PCB/BeagleBone_Green_Wireless_V1.0_SCH.pdf
It adds a WiFi and Bluetooth module which uses a number of signals which appear on the expansion header, and which were not used by previous generations of the board. As far as I can tell, this is not well documented, but compare p. 9 to p. 10 to see which pins are in use.
In particular, it uses the same four McASP (audio) pins that the Bela cape uses, as well as some of the GPIO pins on P8. The ADC problem you see would seem to come from the SPI0 pins, not McASP, and those pins aren't affected as far as I can tell. So this is only a partial explanation.
Still, for any of these high-speed traces, extra load on the board could create signal quality problems, with unpredictable results.
As it happens, the BBG Wireless uses GPIO1_29 as the OE (output enable) signal for the level shifters which drive the WiFi/BT module. That's Pin 26 on P8, previously unused by the BBB and BBG. It's also the same signal we use for speaker amplifier muting on the Bela cape. So unmuting the amps also means enabling all those bidirectional level shifters.
A quick and dirty test would be to run your program with the -M command line argument (mute speakers), which should keep those level shifters turned off. Have a look/listen to all three major converters -- audio, analog in, analog out -- to see if there is any change in result on them.
From there, a more detailed investigation would involve checking the signal quality on the high-speed traces on a scope. I will also see if I can get myself a BBG Wireless for testing.