Gaming would be worse off. Period.
Not only would we be without multiplayer in shooters (which can be darn fun), but we'd be without multiplayer in racers, and RTS games, and various other games. Multiplayer provides a lot of replayability to a game (not to say that single-player can't have good replay value, I've replayed many a single-player campaign in my years of gaming).
If you had a friend who lived hundreds or thousands of miles away, you wouldn't be able to play games with him/her very often.
MMOs would not exist. At all. No WOW, no Rift, no SWTOR, no Everquest Next, no EVE, no Dust 514. None. Think about that...an entire genre, one that has generated many billions of dollars for various developers (mainly Blizzard, mind you), would simply never have existed.
There wouldn't be MOBAs either.
Even single-player games would be worse off. Designers learn from each other; they see the successes that other devs have had in other fields, and apply them where possible to their own work. Dark Souls, for example, would be a lesser experience if it did not have the online features that it does.
Missing out on all of that would be terrible. Not because everyone likes every single thing I've mentioned above; I'm sure there are some people out there that don't care about ANY of the things I've listed. But gaming isn't about the specific desires of any particular niche group. Gaming is and should be a growing and diverse medium that has experiences for people of all different stripes. This brings more people into the industry, leading to larger install bases, which in turn allow developers to make more money and make bigger and better games.
Does the growth associated with online games lead to same growing pains? Of course. Because the industry has grown and is far less isolated from the rest of the world than it once was, we get put on the hot seat more often for tragic events that have little to no connection to our medium. And because we as gamers are less isolated from one another (since we play multiplayer games online together), we naturally encounter more of the ugliness present in human nature than we otherwise would if our multiplayer experiences were still confined to the couches in our living rooms. But these new challenges are a chance for us to grow as people and as an industry. We find new ways to promote good behavior online, and we learn how to better respect and interact with other people.
There is plenty of space in the industry for single-player games and games/game modes that can be played offline. Just because some offline/SP-heavy genres aren't as dominant as they used to be doesn't mean they are going to just disappear, and anyone who claims that gaming would be in some wonderful place if developers would only pander to their specific tastes needs to take a step back and take a look at how much richer the industry is because of the variety we see in games today.
