Agents can save their reasoning into markdown files, and commit those files to Git. Are "Checkpoints" just a marketing term for that, or there's more to it?

Claude Code already does this, you can access it with /resume, /rewind and /fork. I'd imagine building a version that saves in the repo instead of in the home folder would take very minimal effort.

This is about doing it seamlessly and flawlessly then sharing it across a team.

So using something like the compound engineering plugin and committing its "brainstorms", plans, and "solutions"?

No, it's not. I can't get leadership's buyin on a Claude Plugin.