There is a lot of crossover between FBA, and a UPS or FedEx, but it's not 100%. FBA is basically B2C, as opposed to B2B or C2C.
Back during the "peak lockdown" of April/May 2020, a little birdie told me about some of the steps that Amazon was taking to give the warehouses and delivery network a break from the increased demand. In other words, Amazon saw lots of unexpected growth due to the pandemic. It wouldn't surprise me if plans to launch "Amazon Logistics" or whatever as a separate line of business got pushed back due to the pandemic.
Edit: I know for a fact that the leadership of the Logistics part of Amazon has expressed a desire to become the "4th flywheel" of Amazon, which is Amazon-speak for being a self sustaining, revenue generating line of business. But as you allude to, they have to soak up all the demand generated from the Amazon consumer business first.