Messages don't delete across devices on iOS 15.1 and MacOS Monterey 12.0.1

When I mark messages as "read", that status change DOES sync across all my devices. However, when I delete an iMessage thread or SMS thread, the deletion only takes effect on the device where it is performed. The deletes DO NOT sync across devices. I have 4 devices:

Laptop: MacOS 12.0.1 (21A559)

Desktop: MacOS 12.0.1 (21A559)

iPhone: iOS 15.1 (19B74)

iPad: iPad OS 15.1 (19B74)

All of the devices have the "Messages" option enabled in the iCloud settings.

Best I can tell, this stopped working when I received my new iPhone 13 Pro Max. I'm wondering if it is because I setup my new iPhone 13 from a "backup" of my old phone. The reason I say that, is because when I navigate to the following on my iPhone:

Settings > Messages > Text Message Forwarding

I see all 4 devices in there, and they are all enabled. However, the entry for the iPhone still has the old iPhone's device name (e.g. "Adam's iPhone 12 Pro Max" instead of the new name I gave the device after the setup. It was either after the iPhone 13 upgrade, or after the upgrade to macOS Monterey.