Sure, squats strengthen your core, but the core isn't a set of magical muscles that automatically make you faster. Strong quads, hams, and glutes are more important. Squats train your legs/posterior chain/core (and upper body to an extent), and also teach the triple extension [0] which is key in explosiveness. Type I and Type II muscle fibers matter as well
[0] http://myweb.facstaff.wwu.edu/chalmers/PDFs/Triple%20Extensi...
Why did you just refer to core stability as "magic muscles" in a disparaging way? What exactly was your goal there?
No one here said your core muscles were the only muscles involved in these exercises, but they are critical.
Stop being a jackass.
https://www.google.com/webhp?ion=1&espv=2&ie=UTF-8#q=what%20...
"quickness" is absolutely about the core muscles. This sort of quickness is not quite what the article was talking about, but for your case it's relevant.
A major reason why squats are so beneficial is because it strengthens your core.