I don’t know much about large western FPS franchises, but, correct me if I’m wrong, isn’t the online multiplayer the most significant part of most modern shooters? As such, wouldn’t a generalized online multiplayer player character that uses the game’s most notable weapons work best?
It pretty much is the largest segment of the FPS playerbase - for example,
30 million played Warzone in early 2020 - and has become the focal point of Activision / EA's efforts with those two games. While there's a valid argument to be made that the overly strong focus on multiplayer combined with the yearly releases has hurt both franchises' legitimacy, the multiplayer remains a major league force in the video game industry period.
That said, I'd argue one could hypotheoretically rep a FPS shooter with its campaign's main character and incorporate some aspects of the multiplayer into that. Those usually recieve some spotlight (albeit not as much as the multiplayer). CoD also has the regular Zombie mode, and it's been brought up a couple times around these forums. The most recognizable campaign characters are Captain John Price (CoD: MW) and Sergeant Henry Blackburn (Battlefield 3). They're pretty much the focal points of the most famous and impactful games.
One of the problems surrounding those two though are their direct ties to real military organizations. Price is a member of the British SAS, Blackburn is a Marine. Now,
is a full-on Black Op and initially a Green Beret, but his main affilitration is with fictional Black Op group FOXHOUND or just saying effectively "**** it, I'm going solo for this mission!". Nintendo might be keen on avoiding any PR headaches with say Mario getting shot at by a very realistic Marine, so a generic multiplayer avatar might be much more attractive because it'd draw relatively less controversy. It wouldn't remove the headache of CoD having ties to the US Army entirely (one reason to doubt it happening), but if Nintendo wanted to I don't think it'd stop them entirely.
Besides, the multiplayer has a ****ton of different loadouts and options (one reason it's remained insanely popular consistently), so there's a strong possibilty that a moveset could be implemented with or without any campaign stuff.
I think the problem with the "celebration of gaming" aspect is how some huge franchise would be super hard, if not impossible to represent properly. For example, huge western shooter like Call of duty of Batlefield. How do we represent that? What characters represent those franchise? Need for speed is the biggest racing franchise (maybe after Mario kart), how would you even want to represent it? Guitar hero, Rock band, Just dance, Dance dance revolution are probably the most popular rhythm games franchise ever, yet people are more talking about Rhythm heaven and Taiko no tatsujin than those. As much as the "ultimate celebration of gaming" would be awesome, I think it could just never work for a lot of big franchise.
CoD and Battlefield do have options, but it's games like say Guitar Hero / Just Dance etc that have very little room for character selection. Partly because they've usually relied on real life star power to help sell their games. For example, Guitar Hero III's cover:
The two on the side are fictional... but anyone who's even remotely familiar with rock music from the late 80s onwards will know who's front and center.
Slash. It doesn't automatically disqualify GH, but it heavily complicates things when real people take front and center of the franchise. Music tracks etc. could hypotheoretically work, but it's a far cry from Rhythm Heaven or Taiko who actually have fictional characters who are front and center.
Gran Turismo and Forza fall on similar premises: The appeal of those two games are to play around with real vehicles on real tracks (such as Monaco, Silverstone or Laguna Seca).
And, there's an argument to be made that Gran Turismo - in its PS1 heyday - were as iconic as Cloud ****ing Strife, and more iconic than Crash. The first Gran Turismo game is the only PS1 game to outsell FF7, and did so barely. Forza for its part has given Microsoft a much needed IP that is high-profile and help drive Xbox sales.
But, the real aspect severely limits if not eliminates the possibility of a playable character from those racing series. Both GT and Forza heavily push the shiny new car(s), and intentionally so - after all, anything new and shiny will be more inviting to a lot of people. That would complicate Smash Bros. negotiations right from the start - Sakurai noted that getting Brosnan's likeness would be too much of a headache, and he wasn't replaced as Bond until 2006. Cars are replaced on the GT / Forza covers more often, and the same goes for the real life musicians who usually sign on for a game sponsorship (Helloooooo "The Beatles Rock Band"!) and not much else.
Still, if Nintendo were hypotheoretically willing to pay the expenses they could go for such a stage or music. That's asking quite a bit, yes, but it's not entirely out of the question. "Only" almost out of the question given how fiscally conservative Nintendo are.
So yeah, there are neccessarily genres which will probably never get playable reps in Smash Bros. Not as long as the current "No 4th parties" rule holds and / or Nintendo continus to be very vary re: just throwing money around willy-nilly.