That'd gate the clock on the bus talking to the digital components, but there is also going to be a separate clock that the ADC will be using to keep track of sampling with less clock jitter. If you start gating that clock off you can end up with some unpredictable results when you start to read back after you turn it back on (unless the IC was built to expect it). This would all have to happen outside of SPI or I2S, which means more hardware and more cost, and probably a more expensive ADC. None of the ones I've seen (been wanting to play with a few) explicitly support doing that because they've got a clock multiplier internally that they use.
The MUX you link to would likely allow them to disconnect the microphone easily, BUT it wouldn't prevent the noise on it's own in the first place. There'd likely still be a signal at 24MHZ that shows just blank audio, it'd be better for security of course but doesn't fix the leakage.