I've been running Docker on WSL 2 for a while now, without systemd or Docker Desktop. There's no need to wait for systemd.

You can install Docker following their Linux guide and then in your shell's profile file add:

    if grep -q "microsoft" /proc/version &>/dev/null; then
        if service docker status 2>&1 | grep -q "is not running"; then
            wsl.exe --distribution "${WSL_DISTRO_NAME}" --user root \
                --exec /usr/sbin/service docker start >/dev/null 2>&1
        fi
    fi

Basically the first time you open a terminal it'll pause for 5 seconds while the Docker daemon is started but it'll stay started for the duration of your Windows uptime even if you close your terminal. The next time you open a terminal it'll be instant.