By this I take it you are talking about shipping a merchant-fulfilled order to a buyer?

It does not matter what courier you use as long as it as an accepted courier (ie tracking will "count"). And the service you select gets it there within the delivery time frame.

If you use Amazon buy-shipping (and tracking shows it was shipped on time), then Amazon has declared they will protect you from claims of non-receipt. (The sometimes renege on that promise)