Time machine is NOT excluding downloads
Great details - since you are in the command line already we should be able to get some very critical details sorted and help you understand how the system works, validate it’s configured and perhaps let you solve this.
The three big commands are tmutil isexcluded and tmutil startbackup --block and tmutil compare
You should be able to check at any time if the file you have is excluded - so start with that and make sure you are agreeing with whether it’s included and excluded as you move it from folder to folder - from ones that are and are not excluded.
Then once you see it included, run the compare against the last backup - see that it’s flagged as different, then force a backup and re-check the compare.

Once you’re sure the backups are happening in the command line (or not) then we can look if you need to diagnose Time Machine itself or if the GUI may be lagging or broken or it’s just a misconfiguration  that can be sorted.