Transferring Data between iPhones

I purchased the iPhone 15 Pro Max recently (upgrading from an iPhone 12) and since I don't use iCloud very much, I always opt to transfer my data directly through the previous iPhone instead of through iCloud. This took around 2 hours to do, but after it was done I noticed some inconsistencies between the two devices.

For example, I use the search bar a lot in the iMessage app and I noticed that my 15 is not showing any of my old saved messages, pins, links, etc like my 12 would show. Another example, I use Find My a lot with my family and on my 15 I can see one person's location, but the rest are failing to locate, while on my 12 I see everyone's location.

For clarification, both iPhones are up to date running 17.4.1 and both are also still logged into my Apple ID (I don't quite want to erase my 12 before I know for a fact I have everything the same on my 15). I think I saw that if two phones are logged into your Apple ID (when in the process of transferring) there is some issues and/or lag with the data. Weirdly enough I did notice that some things did start to pop up on my 15 after a while, but not everything.

So my question is: After completing the data transfer from my 12 to my 15, does the new iPhone still need time to download/process some things in the background after saying it's complete? I do vaguely remember this was the case when I transferred from the XS Max to the 12. I just want to make sure I'm not losing any of my data before factory resetting my 12. Thank you