USPS does run on Saturday.
USPS post offices are open on Saturday and USPS mail carriers do deliver on Saturday as well.
Saturday is considered to be a business day for USPS.
Yes there is still mail delivery on Saturdays.
I get my USPS mail delivered to my curbside mailbox on Saturdays all the time so at least here they are still delivering USPS mail on Saturdays.
However some post offices have limited hours on Saturday.
For example the post office here in my small town is only open for 1 hour on Saturday.
However the mail is delivered to addresses past that hour the post office is open.
Mail will continue being delivered on Saturday for a long time from now because the USPS wants to and needs to get packages and mail delivered so that it does not back up on them.
On Sunday in some locations I've seen USPS out delivering mail but it's usually in larger cities and larger towns that, that happens in.