Yes USPS does deliver all regular mail and packages or any mail that is coming through their system.
If you're expecting a package or other mail on Saturday then as long as the mail or package has arrived at your local post office on Saturday then the USPS will either deliver it to your mail box or to your house depending on how you get your mail.
The USPS here in this small town is only open a few hours on Saturday but I do get mail on Saturday although I don't always check my USPS mail unless I'm expecting a package.
USPS just doesn't deliver mail on Sunday but on Saturdays they do deliver mail to addresses and to your post office box.
But even Sunday and all other days USPS is delivering mail through their distribution centers and on the semi trucks and airplanes 24/7 so that the mail gets to you on time.