Bit of SNL trivia: When it comes to the weekly sketches, who is cast in each skit is up to the writer of it, so if, say, Aidy Bryant gets a sketch on air, she'll probably cast her friends in it. Hence why you see a lot of sketches with people who are friends in real life in them.
You didn't see Pete Davidson in any sketches because the cast didn't want you to. It's obvious that the cast felt abandoned by Pete, it's his very last episode and he didn't get cast in a single sketch. Not even pre-recorded content. The "Prom Night" sketch in particular had every single cast member except Pete.
It's telling that Pete's only bit the entire episode was a 3 minute weekend update segment, care of one of his only 2 friends left on the show, Colin (the other being Lorne Michaels). Notice how Pete only thanked Lorne? Not the cast he worked with for multiple years, or writers, or crew who put up with him.
A 3 minute whine fest, a very fitting end to a very lackluster SNL career.