Question: Why isn't Andy Samberg in lots of skits?
Answer: It actually makes sense why he isn't in lots of skits. Andy doesn't write skits for himself, he doesn't work with the other writers to make skits for him, and he doesn't even ask to be used in skits. He used to try to be involved a lot more, for the first few years. Now he just relies on the digital shorts.
He isn't really motivated to be in more than one skit a week, since he's paid per episode, not per skit (and being in more skits just knocks other people out). So he was paid the same for being in the one skit this week, Bidet, as he's paid for episodes where he's in 6 or 7 skits.
So he just appears in skits when it makes sense. He's certainly not trying to "skate by" or "be a slacker." But if the hosts and writers aren't requesting him, then pushing himself in just pushes other people out. I think Jorma realized this as well, which is why Jorma eventually gave up trying to be on camera a lot.
Andy Samberg as the Swedish Chef:
Also, because Andy never really pushed through very many characters or impressions (other than the Swedish Chef, which got 3 sequels), Blizzard Man (which got 2 sequels and one non-SNL appearance), and Cathy (which got 2 sequels), he isn't in the writers' minds (because they don't see the range of characters that he can do, at least not in comparison to the other players).
Andy Samberg as Diablo Cody:
So it also seems that he is occasionally tapped by the writers to do impressions that make sense (I'm not sure how many of these Andy thought of himself, but a lot of these were likely given to him by the writers): Rahm Emanuel, Zac Efron, Sanjaya, Nicholas Cage, Kevin Federline, Dustin Diamond, Jack Johnson, Diablo Cody, Billy Bob Thornton, etc. It's also likely that some of these came because writers wrote parts for these impressions, and then Andy was like, "I might be able to do that. I'm going to go practice my impression." And then he comes back and "auditions" for the role.
Andy Samberg as Dustin Diamond:
Sometimes there are hosts who are very comfortable with Andy, and he ends up in lots of skits that week.
Follow up question: If Andy isn't motivated, why doesn't he just leave SNL?
Well, it isn't that he isn't motivated to be on SNL. It's more that he isn't motivated to kick the other players to the curb and take over SNL. It's just a perspective shift. Why work hard to be the star and push other people out of the limelight who maybe are better at characters and impressions?
Flags of the World:
Andy certainly embraces characters and impressions whenever he thinks of them or the writers think of them. And lately he's been owning more SNL Digital Shorts (doing both the acting and the music more than he used to). For example, he recorded his own voice for The Tizzle Wizzle Show and Flags of the World.
Also, this year he's been concentrating on doing more original characters in the digital shorts, such as On the Ground, Shy Ronnie, and Bootie Call.
So he's certainly motivated to continue "what works" for him on SNL.
Also, it seems that Andy's contract will be up next year, his sixth season. So that year may be his last. (You can always tell how many seasons Andy Samberg has been on by how many Laser Cats there are.) So if the question is, "Why doesn't Andy just leave?"... then that's also a contributing factor. He wants to honor his contract.
Also, what is he going to do once he leaves? I don't think he currently has an answer to that question. We have ideas for him, but Jorma and Akiva are directing movies (so a TLI show might not fly), and he hasn't been able to break into the movie business (yet). Compare that to Bill Hader and Kristen Wiig, who are sought-after supporting actors.