So I'm going to mark this one as [SOLVED] as what I posted above did seem to fix it. I'm still looking into my build more now to see if anything else might have been broken along the way, but for now, so far so good! Here's what I did.
I put my current SD card with the older OS into an older Pi 4.
From Emulation Station I pressed F4 to get into the terminal and typed:
sudo apt update && sudo apt dist-upgrade
The update took about 40 minutes to complete.
Once I did that, I typed "emulationstation" into the terminal, and did a system shutdown from ES.
I then re-booted the now updated SD card to make sure it worked in the old Pi, and once it loaded up, I shut down from ES once again.
Took that updated SD card and put it into the new Pi 4 and it seemed to work just fine.
Immediately made a backup of that now working SD card! <-- VERY IMPORTANT!!!
The SD card now seems to work just fine in both the old Pi 4 and the new Pi 4.
I still don't quite understand why the old SD card won't work in the newer Pi 4's as I'm still using Debian Buster on both versions of the SD card. I also had no idea that there was any OS on the actual Pi board itself as I thought it was all contained within the SD card. I'm not exactly sure why my build wasn't working on the newer boards, but I'm at least happy I've found this work around.