The ability to click down and then drag on this green dot button in order to start a split screen view was removed in 10.15 Catalina. Apple removed it in order to enable sending to Sidecar. However, I don't think it's very intuitive for users trying to use split screen as it is a definite slowdown. It doesn't feel very "Apple" to have to pause and squinty-eye a menu instead of just point where you want it to go. I know Toolbox has a tool that lets you use a sort of "Aero-snap" like feature. I don't personally use that (I have Rectangle), but I would use the heck out of this (I haven't used split screen much at all since this change Apple made, because it's smoother in my own experience just to four-finger swipe.)