I realized that in iCloud it was showing that Messages was only taking up roughly 1GB of storage (which seemed incredibly low). On my iPad (which had all the data), I toggled the Messages toggle off then on under the iCloud sync settings. The next morning this caused the iCloud Messages storage to increase to about 6GB. Which seemed like an improvement, but still lower than I expected (considering that local storage was like 20GB for messages). This also lead to some images showing as "Download Image" (I forget the exact text) buttons in the conversation instead of empty chat bubbles, and tapping that downloaded and displayed the image. However some images were still empty chat bubbles.
Today, I erased the new phone, and set it up using the Direct Transfer function instead of iCloud Backup Restore. The local storage for iMessages now shows at roughly 20GB (which is expected), and after a few tests, it seems like images are showing up. Sadly I can't completely verify that the images I knew weren't syncing are there until the iMessage Search function finishes indexing my messages. I will try to report back after this is done to let everyone know if this worked.
It seems like this is some issue with really poor iCloud syncing behavior. Not sure why, but really seems like another Apple bug... 🙁