Time Machine not detecting changes in certain folders
What actually helped was to identify the folder inside which no file changes were detected, and then rename that folder and force a backup. Then all new content inside it got added to the old backup. After that, I renamed the folder back to its original name, added another file inside and did another backup - again, the new file got backed up.

So, somehow, that folder got stuck, and by renaming it, I could reset the backup for it.

Still, I will keep an eye out for other folders that may not get backed up, using the tmutil compare command regularly.

You just can't trust Time Machine.

That's why I also use CrashPlan as a secondary, and offline, backup.