Thank you for confirming that deletion is occurring for you. I will raise a ticket with our environment support team!
Please forget about Cost Category. It has nothing to do with this. I had just placed a simple table there, populated it with an aggregate, and added a delete button. Since there's no logic to create a list, the save action that I'm using with the Budget Type screen will not work, since there would be no list sent to the server action.
Since you confirmed for me that this works fine in your environment I'm marking this as answered!