So here is what I did yesterday and it was by far my best SmartSwitch experience/transfer ever:
1. Switched my SIM card into my S22 Ultra immediately, from my S21 Ultra, before turning on the S22 Ultra.
2. Used the cable that came in the box of the S22 Ultra to do the transfer, which seemed thicker than most of the Samsung cables I've had (I'm assuming this is a good thing).
3. Went through the initial process of connecting to Wi-Fi on my S22 Ultra.
4. Then I went right to the SmartSwitch transfer on both devices. I chose to transfer all.
5. EVERYTHING transferred over for me with the exception of my bookmarks on the Samsung internet browser. The apps on my home screen transferred over perfectly, even the phone remembering my saved Wi-Fi networks.