I have a 2019 Mac Book Pro running Big Sur 11.2.3 and I have Better Touch Tool set to launch at login both in my computer's system preferences AND in BTT software preferences, and yet the window snapping part of BTT does not launch at start up when I restart my computer.
I have this problem too. Also when I wake up my Mac after it goes to sleep, BetterTouchTool has quit and I need to open it up manually. Both these features have check boxes in the settings window and I have both features switched on. I have a 2019 MacBook Pro running macOS Big Sur v11.2.3.