I don't think it's about "first-person shooters". Nintendo has produced first person shooters before with Goldeneye and Metroid Prime. While they had relative success, neither could save the N64 or Gamecube.
The real reason games like Call of Duty and Halo are successful is because they offer the definitive online experience. The fact that they happen to be FPS is advantageous, but not the core element of their success.
The problem, I think, is Nintendo's odd approach to online multiplayer. While the Wii had potential to go online with relative ease, they decided to incorporate a complicated code system that made both developers and customers reluctant to try online content.
If they wanted to fix this, they would have done so by now. Unfortunately I think Nintendo has decided to ignore online gameplay of any kind.
If I was an optimistic person, I would say that Nintendo might probably release a "Wii Sports Online" within the next year (paintball included) and completely revamp the online system undoing friend codes permenently.
Being a realistic person, however, makes me believe that Miyamoto wants to inflict Wii Music Plus on us with a UGC-filled Pikmin 3 not far from it (no online gameplay in either of these, of course).
