This configuration should indeed be very straight-forward… I’m also assuming you’re running Docker Desktop for Windows? What version of it are you running?
Part of me is wondering if you’ve tried to do a apt install docker.io directly in your WSL2 distro (you do NOT want to do it this way). If so, I’d try wiping your WSL2 distro, install a fresh one (Ubuntu is fine, btw), then install Docker Desktop for Windows. Then you’ll have the full integrations for Docker across Windows and WSL2.