Transfer iMessage Archive to new Macbook
Since your message archive didn’t migrate the first time, I would do this:


Connect a hard drive to the old mac and let Time Machine make one backup - you can configure it to skip large files if you have lots of videos and photos you know can transfer, and you can delete any large apps you know you won’t want to back up or migrate - so light cleaning before one backup.
Then - see if you can update the old OS - I would update it to the absolute latest since that will do two things. First is it runs migration to make the older data ready for newer OS. Second, it makes the migration assistant newer so you’re not bringing data from 7 years back OS to the latest.
Erase your new Mac (back up any files needed) and re-run the migration from the updated Mac or a second backup of the upgraded Mac if you want.


I use the PhoneView app to manage messages on macOS so that might be an option if Apple can’t update and migrate the data with their normal tools.


http://www.ecamm.com/mac/phoneview/


It’s targeted at managing messages from macOS, but the free trial might show you if it will help manage / archive your messages in a meaningful way on the mac side. I’m sure someone knows forensically how to parse the database files, but that needs some special work based on your exact macOS version. Maybe someone with that skill will see this and answer how to directly open the files from ~/Library on the old mac and then transform them / process it to load from the newer macOS app.