Items from an Exchange Online System folder called "Files" are appearing in Outlook's Instant Search

The Files folder should not be exposed in clients, it's a system folder used to store copies of documents/attachments. You can read more about it in Tony's ramblings here: https://www.petri.com/mysterious-files-folder

One of the things you should do and I don't see listed above is to make sure you're using the latest Office build, as if I remember correctly some of the issues around the Files folder required client-side fixes. If you feel comfortable working with tools such as MFCMAPI, you can also explore one of the affected mailboxes with it and check the properties of the Files folder. As the Files folder only exists for O365 mailboxes, I dont exclude the possibility that the migration tool didn't know how to process it and/or made some changes in its properties. But that's just speculation on my side, best check with the vendor support, and also O365 support.

Another thing that comes to mind is to move the mailbox to a different DB and see if there are any changes, you can do this by running New-MoveRequest in the service.