I have checked it. All payments from the DHF to the HBD stabilizer go back to the DHF. Plus profit.

If not, why not reduce it? Why is e.g. not 6000 per day enough?

According to this comment, this proposal would not even be necessary if more users with their own coins worked the way the HBD stabilizer does. And they would even make a profit.
I would like it better if the Hive users would secure a stable HBD on their own initiative. But as long as this does not happen, I will support the central protection by the HBD stabilizer.