By singleplayer, we don't always mean the campaign. Usually it just means content that can be played by anyone, alone.
When you buy a game, you expect to be able to access and enjoy every part of it. In no way is this an unreasonable expectation. You don't buy books expecting a chapter to be missing, or a film for the same reason. Singleplayer campaigns can and are able to be experienced by anyone who owns the game. Preventing someone from accessing arena instant action because
a) they only have one controller
b)do not have a reliable internet connection
c)haven't the disposable income or time to devote to online subcription services
d)have nobody who wants to play that particular game with them
e)or all of the above
is simply not good enough an excuse. It may sound overly dramatic, but this is discrimination, plain and simple. These poor bastards even had to pay the same price for this half-product.
I'm a huge fan of multiplayer, and one of my all time favourites is the original Unreal Tournament, probably the first successful game to be sold just on multiplayer. It had everything; deathmatch, team deathmatch, assault, capture the flag... it was a veritable powerhouse of varied multiplayer gaming. But there's something very important that modern shooters just don't understand, that UT knew all too well- the importance of bots.
You see, no matter how varied the maps, weapons (and later on, the vehicles), the bots could always go toe to toe with players and could make effective use of anything the player could use, too. This is what is key to UT's brilliance: every single game mode, map, weapon and powerup could be used by, alongside, and against the player by the AI- preserving every aspect of gameplay for players that play alone
In this way, a game can have no story based singleplayer and still work well. Look at Mario Kart or Mashed. Great multiplayer gameplay, zero story, and yet compelling for solo gamers because there is AI to carry it over (unless you want battle mode, in which case substitute Mario Kart for Diddy Kong Racing). For games doing it the other way around, only those playing online get all the content. Everyone else is left out in the cold.
And speaking of getting left out in the cold, think about this: There will always be a finite number of players, and hours in the day. With more and more games competing for online multiplayer time, chances are that over time, a game you particularly like will have fewer and fewer people playing it, and soon the multiplayer will dry up completely. If you see games as disposable distractions until the next big thing to spend your cash on -as the publishers no doubt want you to think- then this isn't a problem.
But for games that stand the test of time, Multiplayer will be wasted unless it can be played alone as well. Look at Tribes 2. For all the fans it still has, most have resigned themselves to never being able to have a big match ever again- it was online only and just doesn't have the support it once did. Bots and challenges give a game an infinite lifespan. Nobody wants to play Goldeneye multiplayer because they're too busy with COD and Starcraft 2, which is perfectly fine, but I don't mind because my Perfect Dark bots are ready to fight, whenever and however I choose to. I for one will still be playing PD multiplayer long after many samey multiplayer only shooters have come and gone.
And look at Deus Ex. It actually had a multiplayer, but nobody remembers that. The thing that brings you back to a good game many years on is not the multiplayer, but the singleplayer. Like a good movie, if the singleplayer of a game is good enough, people can and will want to reinstall it and play it again every so often. This never happens with multiplayer except for the abolute peak popular ones, namely Counter Strike 1.6. But sending off a game without bots just hoping it will be the next counter strike is incredibly naive. there is NO excuse not to have multiplayer AI capable of all the game modes and weapons/vehicles as the player.