Problem: The data of a list wasn't exported when I saved the list as a template with content.
Context: I created a list using a Feature with schema.xml and manifests for list instance and list template. My custom fields were not showing up in the list settings under columns. And data wasn't exported.
Cause: I had my fields set as read-only in schema.xml (Fields/Field elements).
Solution: I removed the attribute and it worked. I had set them to read-only for performance in the first place. I don't think it makes much of a difference anyways.