I pit headphone manufacturers...

There are set ups that allow you to plug a Bluetooth transmitter into a 3.5mm and pair your headphones with that, but I’m not sure about whether or not that’d be permitted on an airplane. (Apologies if you were asking about the second part rather than logistics–I have no answer on that topic.)

And really, a $20 set of headphones ain’t all that…

That’s inherent in any self-keying microphone. Voice-operated switch - Wikipedia. It has to hear sounds that it recognizes as voice, not background, before it turns on the rest of the system.

The fix is to understand that and behave accordingly.

Start every utterance with a deliberate voiced full-volume “uh” and it’ll be fine. Whether you’re talking to ATC or chatting with your cockpit-mates. You’ll sometime hear other pilots on the radio who’re habituated that way even though they’re using a conventional PTT microphone. “Uh Center, this is ABC123. Request blah blah blah” … “Uh roger, descend …”. Where “uh” isn’t some absent-minded verbal fluff, but a deliberate word meant to key the mike.

Having to say “Siri:” or “Alexis:” before every command is the same idea but vastly smarter. Aircraft headphones are dumb; they’re just listening for noise different from the ordinary drone.

Amateur radio operators using Vox (voice operated trsnsmitter) have the same issue/habit. I generally use a footswitch , but that’s not practical in many instances.