Here are some places an Auto-Squelch feature could be added:

In the options menu, where there is a lot of blank space and the option is easily discoverable by players looking for it. But then we'd need an extra button to improve the player experience at no cost to others. Ouch

If we're deathly afraid of adding a new button because that messes up the UI, we can change the function of the existing button to toggle emotes on/off for all players and just not reset each game. But that might confuse people too used to the old system...

A second "Squelch All" On/Off toggle could be added in the blank space next to the existing squelch. While this is another button, it doesn't take up additional UI space or change existing functionality. Very mobile friendly and discoverable with current squelch

So there could even be a "hidden" command, like there is for streamer mode. You might not know this, but by clicking "Ctrl+Shift+S" you appear as "Your opponent" instead of your name. It's less discoverable, but it takes up no UI space whatsoever.

There are few arguments I see against this feature. The button argument fails because there are options that involve no additional buttons. It causes no players any harm while relieving frustration from others that doesn't need to exist

It's been requested for about 7 years now. It's an important feature, as evidenced by the squelch feature already being deemed valuable enough to be in the game in the first place.

Let's just not need to click constantly if emotes are distracting or unpleasant