I think "over-saturation of first person shooters" is an over-saturated statement, considering the variety within that designation. People tend to lump everything "FPS" together even when those games are not in the same genre or have key differences.
On topic, I think Nintendo wants the visual of Mario on the screen most of the time. They are very zealous about protecting their primary brand since they lean so heavily on it.
However, I could see some type of game where you "help" Mario as a different character in FPP, perhaps in a Slime Rancher type environment.
I think a Mario game where you can't see Mario is bad for branding. Just a red sleeve and white glove holding a gun... yeah I can't say that would be smart. It's a neat idea, but not for Nintendo - they're market leaders, not followers.
They're clearly both, even if the following part is at a slow and deliberate pace.