For push to talk to work, you need to have access to keys even when you're not in focus.
That's not something doable today on the web for obvious security reasons, but it's possible for Discord that has a separate app, would be doable for Zoom too I guess.
Interesting sidenote: PTT works fairly well on mobile. I'm in a lot of meetings where folks are using their computers for video and "dial in" for audio on mobile so that they can continue working and then PTT on the phone which is now functionally a giant dedicated button for speaking.
