Many of modern aircraft is supersonic so acoustic detection would be useless, even if it technically works.

True (though stealth bombers are subsonic, just). What about headings that are not straight at you though? May still have value detecting things passing by?

When I was at military I attended a few exercises where jets were involved. They don’t go supersonic but pretty close to it so usually once you start hearing something it’s almost over. It’s actually quite terrifying to have a jet coming at you at 800 km/h at low altitude. Very surprising, then very loud and incredibly fast. You also hear passing by jets only once they are long gone. I assume radar is just much better than sound.

Not just jets, a regular propeller based plane can have that effect:

https://www.youtube.com/watch?v=4iOoiEbtf2w

Now double or triple the speed, make it much louder and you have a jet.

Most do not have supercruise, so they would spend most of their flight at high subsonic speeds.

Then it becomes a battle between the detection range of the audio sensors vs how long the attacking craft can spend supersonic. They could spend the last phase of the approach supersonic and even transition then turn to make any tracking useless.

If they are supersonic, would you just need to increase how much you lead the weapon. Supersonic doesn't mean they're quiet, they're just further ahead than where you are hearing them. Or am I just totally making stuff up?

I think much of the usefulness is tied up into the sound arriving substantially before the plane itself.

Or you have a network of sensors?

Detecting helicopters would be one situation where it could be useful for mobile troops on the ground.