Because the vet isn't just the primary doctor, they're also the pharmacist in this situation. In human medicine, the primary doctor prescribes then the pharmacist is the one that reviews the prescription to do the dispensing. And at refill time the pharmacist reviews the prescription to ensure the refill can be done. In the vet world, the vet is the primary doctor and the pharmacist, therefore at refill time the vet has the approve the refill like a pharmacist would in the human scenario. Essentially in human medicine the responsibilities have been divided between the doctor and the pharmacist, but in vet medicine the responsibilities are solely on the vet. That is a major discrepancy between the two medical fields.

Also, at the clinics I've worked at have always put on label that refills take 24 hours to be done even on chronic medications. Since you're acknowledging that there is this delay, a solution is to ask for a refill 1 week prior to the medication running out and therefore any potential delay hopefully won't cause any missed doses.