Yes the USPS mail does get processed on weekends.
When you mail something on Saturday the mail will still be picked up by around 4:00 PM to 4:30 PM depending on your location.
But on Sunday mail does not get picked up from the post office so if you mail something at your post office on Sunday or after 4:30 PM on Saturday then the mail will sit until Monday.
But if you mail something before 4:24 PM then it will get picked up and sent to a sorting facility where it will still be processed.
The USPS mail is always being processed 24 hours per day 7 days per week on weekdays as well as Saturday and Sunday.
The USPS sorting facilities are always open no matter what day it is or what time as they never close.
It's just the local post offices that close on most Holidays and Sunday.
But mail is always being processed so that the mail gets delivered to the addresses on time.