I'm not sure if this is a bug or just a strange behavior. In version 24.1.31 when a patient has an unsent statement and their clinic is unassigned the statement will not show up in the billing screen. When you exit the billing screen and it asks you to delete all unsent statements, it will not delete those statements that have an unassigned clinic because it can't see them. The next time you open billing it just opens the list of bills with nothing in it. There is no way to run a new billing filter and no other way to see what bills are unsent. You need a custom query to find unsent statements. My suggestion would be that when you select 'ALL' clinics in billing, that it includes unassigned clinics.
I know you can prevent new patients from being created without a clinic in the settings, but there are some third party apps that create patients and don't follow those rules. Either way, I think there should be some way to track down those statements.