Before we get into the weeds - yes, using the MT trigger to power off the camera's switchport is perfectly adequate. But it's not preferred.
Is there a way to set up the MT30 trigger to only turn the camera audio to off or on through a webhook or API? It's obviously not in the other small set of preset automations. I've never set up a webhook, but skimming the first grafs give me the impression that i could set up the button to send out a web message to a web server, then . . ? And, no I'm not set up to develop an API.