Only time we have had a problem with ‘late shipment’ is when we hand the parcels over in good time, but the items don’t get scanned until after midnight.
Usually in these instances and the dreaded ‘Drop in On-Time Shipment Rate’ mail arrives, it can be explained as such - that we shipped on time but a small number were scanned late.
As for On-Time Delivery - yes, this is pretty much outside of our (sellers) control. If you can drum up a letter, or even a screenshot of a webchat etc with an admission of fault by RM, that’s usually enough to avoid any action or to get re-instated.
Amazon don’t really seem to care about the shipment problems, they only care about the Customer Experience and the Prime promise. So mentioning “any impacted Customer who opens a ticket for ‘missed fulfilment’ will be offered pre-paid return or discounted or refunded in full”. Usually seems to grease the wheels.