As Mortono correctly pointed out, online play has not been a central focus for Nintendo. Likewise in the article Carl2291 cited fully explains:

"The fact of the matter is we know we create great content for younger consumers, we know we've got great content for more casual players, and we want fantastic content for that more active player who loves Metroid or Zelda but maybe also wants something like a BioShock 2 to play as well.

"And we also recognise that we don't create that type of content ourselves. We're not good at it and it's not a key focus area. So we want that content on our platform, so we have to court third-party developers and encourage them to make [those games], but it also has to be financially viable for them as well." -Reggie Fil-Aime

Nintendo has never made a great 1st party, first person shooter game. Nintendo's best has been Goldenye 007 and Perfect Dark, while Rareware was under a 2nd party contract with them. Since the mid-2000s, Rare has gone from the platinum success of Donkey Kong Country and Goldeneye to a long list of mediocre games, then Viva Pinata.

Lastly, as scottie elaborated, Nintendo just does not need to at this point in the generation where their titles hold a stranglehold on the top 20 best selling games of the 7th generation consoles.

The real question is why should Nintendo invest all the time, money and creative energy into creating a complex OS to accomodate quality, reliable FPS play and  FPS games when they can pump out another cutesy platformer at a fraction of the cost, which will sell double or triple what a COD game would sale?

This really depends on whether the intent is to prove that Nintendo can hire a Western studio to develop a Nintendo published FPS using one of Nintendo's many IPs, or to that Nintendo's 1st party studios in Japan can show everyone how FPS are done or at least do something new with the genre.

If it's the first, there is definitely an untapped market for FPS games on the Wii, albeit smaller than that of PC, PS3 or Xbox. But the FPS genre just won't ever be one of Nintendo's bread and butter platinum sellers. And farming an FPS project out to a foreign studio with substantial experience in FPS games wouldn't really prove much of anything other than the limits of the FPS market potential on the Wii. 

If it's the second, I'm not really convinced that even if they could "show everyone how FPS is done" in a market where companies know the genre so well they have built their success upon nothing but FPS games. At best, they could come up with something different that steps outside the norm for the genre, like Little Big Planet style "paintinator" game play, or even an "FPS" game like Portal.

For one thing, Nintendo is very much a Japanese company that makes games that reflect its roots and FPS games have always been one of the genres that simply never caught on in Japan as they did in other global markets. One could even say that the Japanese just don't "get" the appeal of FPS games.

Any attempt at such a game would really just be an exercise in showing how they can adapt a genre they have almost no experience with into a game with the same broad appeal as say Mario Kart with racing games.