No you cannot pick up a package from the USPS distribution center.
The USPS distribution center is not like your local post office but instead the USPS distribution center is a big building with conveyors and equipment and people in the USPS distribution center are sorting the packages and loading them onto trucks, planes etc.
When the package is at a USPS distribution center it cannot easily be found anyway so you cannot pick it up at the sorting facility.
Once the package is sorted at the USPS distribution facility it leaves that night or next morning and gets to your local post office or to another distribution facility.
When mail goes to a USPS distribution center the mail is sorted the same day and then leaves the sorting facility or distribution center that night or early morning.
When you mail something at your post office it goes to a major distribution center where the mail is then sorted and then the mail either goes out of state to another USPS distribution center closest to the address.
Or if you're in the same state the mail may leave the USPS distribution center that night or morning and then head to your local post office or a post office closest to your address where it will then be delivered.
Mail does not stay in a USPS distribution center longer than one day.
When mail goes to a USPS distribution center it leaves the same night or next morning.
The mail will not stay in the USPS distribution center longer than a day because the mail is then sorted and put onto a truck for shipping.
Thousands of packages and mail are being sorted and shipped daily and the USPS distribution centers stay open on weekends and holidays as well.