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Open System Settings › Privacy & Security › Full Disk Access and turn on Textbound. If it isn't in the list, drag Textbound from your Applications folder into the list. If macOS offers to quit and reopen Textbound, choose that; otherwise Textbound goes on by itself within a second.

Your Mac only has the messages that are on it. Turn on Messages in iCloud on both devices (on the Mac: Messages › Settings › iMessage › Enable Messages in iCloud) and wait for it to sync, then choose File › Reload Messages (⌘R).

Messages keeps some older attachments only in iCloud when Optimize Storage is on. Open the conversation in Messages and scroll to the photo so it downloads, then export again.

Textbound uses the names in Contacts. Add the person to Contacts, or make sure Contacts syncs on this Mac, then choose File › Reload Messages.

Exports show times in your Mac's time zone, and say which on the first page. To use another one — the time zone of a case, for example — choose Textbound › Settings › Times shown in.

On the Mac the export was made on, choose File › Verify an Export… (⇧⌘V) or drop the PDF on the window. Textbound makes the export again from Messages and compares every page.