I have previously worked for USPS and can help clear up what is happening. The mail carrier and their assigned station (Post Office) should be thought of only as the ingress/egress of the postal system. It would be wrong to say that no sorting happens there, but the type of sorting that doesn't happen there is why your mail is being postmarked at a plant (regional center) fifteen miles away.

I'll use the location that I worked at as an example. In Wilmington, NC, once your carrier returns to the station, they drop your letter into a collection bin. This collection bin is driven to Fayetteville, NC, where it is then processed via OCR. After being OCR'd it has a machine-readable bar-code, and will be sorted for delivery to a local station. The machinery involved in the OCR and sorting process is larger than any of the local stations I worked at. Additionally, almost all of the mail pieces that I delivered were intercity, not intracity. Your use case is the exception, not the rule.

There are many exceptions and corner cases to what I described, but that's what the process looks like for most mail pieces. Locating the scanning and sorting machinery at a hub rather than each individual spoke makes sense. UPS, where I currently work, has some minor differences, but the hub and spoke model is largely the same.