Just soldering the existing circuit won't give you more signal but it should be more reliable. Since audio signals can vary a lot it's hard to get "full range" unless you add a potentiometer and adjust the level all of the time. And of course, the pot only reduces the level. So it's up to you if you think it's worth the trouble of adding an amplifier.
You should be able adjust the sensitivity in software. It's also possible to make the sensitivity automatically adjustable. I do that with my effects. Take a look at my World's Simplest Lighting Effect. It's just a simple demonstration effect that flickers-on an LED when the sound is louder than a 20-second moving average and flickers it off when it's below average. That means you get lots of "LED action" with loud or quiet music.
My code was written for an Arduino Uno so you'd probably have to make some small changes. And I don't know if the ESP32 has a built-in LED or if you'd have to add one.