Um, you don't
have to. No one's forcing you to download that extra content. While I do agree that DLC is often used as a cashcow to put out overpriced, mediocre content, it has little to do with how useful the internet is for gaming. Blame companies that put out shitty DLC, not the internet. In itself, the concept is great - you can download additional content for your game from the comfort of your home. It's just that it's often used as a cashcow of sorts.
In any case, I can't think of an instance when I really hated the online component of a game. Not even Diablo 3 was as bad as people made it out to be, and co-op in that game is much more fun than soloing. I'm actually playing Dark Souls these days and being online just makes the experience that much more enjoyable to me. On PC you also have enormous modding communities that are constantly working on new content and patches, sometimes long after the devs themselves have stopped supporting the title. Morrowind got a
complete overhaul made by fans, while a series of patches and bugfixes were released for Vampire: Bloodlines long after the company that made the game went bankrupt. I'm sure there are more examples, but these were just the ones off the top of my head.