I think a large part of why social networks have desperately failed to reproduce the sense of togetherness has something to do with the state of the internet at large.
I used to play AstroMUD in the 90's and grew close enough to the community that when I stopped playing in college and started 5 years later, there were people who still remembered me. For some reason, that sense of community never transferred to the graphical mmorpgs (for me, anyway). Instead, the gaming community grew more and more toxic and I've since cut all ties with that part of the internet. Something similar happened with the progression from IRC to public chatrooms and livejournal to facebook.
The only real pattern I've been able to see in all this is one of inclusiveness. MUD, IRC and (the early years of) livejournal were hard to use for normal people. It artificially restricted the user base. The restricted access increased the value of the interactions therein. The threat of a ban also carried more weight, since getting another IP wasn't as easy in those days as it is today.
The theory is still kind of half baked and I'm not sure if I can focus it into a service that will reward people for maintaining a smaller, more intimate online network instead of the sprawling thing we have today.
I think the pre-social network Internet had a more homogeneou userbase: more text driven, more verbose, more learned, more, shall we say, "hackery". Or whatever. But there was a distinct property to that userbase that you don't get in the mainstream.
I think a lot of it has to do with the bar for entry: "RTFM" requires both reading and a manual. Anyway, half baked theory.
I think once facebook, and the more "real identity" networks came around, that "natural" communities got destroyed. I was thinking about some of the friends I had made when I was 13 that I still occasionally talk to, that I have never seen pictures of or even heard speak, that I had met online.
Back then groups were formed over interests, and the interests itself could be very niche, but the people you met were also interested in what you were interested in. It was like a self selecting group of uber-friends. The only step in relationship building was really "oh you like thing? I like thing too." Relationships then were really about shared-interests.
However once facebook rolled around cyber communities became more like digital archive of real lives. You don't friend someone on facebook because they had a similar interest with you (most of the time), you friend people that you had some in-life experience with. Now internet relationships are now really about shared-experiences.
Another way to look at it is if imagine you went to a bar by yourself, and everyone else in the bar came alone. You might imagine eventually people would group themselves by similar interests (although certain human dynamics might lead people to group with other people visually similar to themselves.)
In the post-facebook era its the same setup, but this time everyone bring 8-12 of their high school friends. Your high school friends may have not been had as much deep rooted interest in AstroMUD, but they are generally all-around ok people. At the end of the day you are really only together because you went to the same high school, and weren't assholes to one another.
If the "feeling of togetherness" is the product, then social networks have succeeded where 2L failed, if we use market acceptance as our measure of success.
It boils down to the question of what makes one feel "together" with another person? With the success of social networks like Facebook and Twitter as proof, many people find a common expression of sentiment typed-out and displayed on a shared-view in a timely manner is sufficient to engender feelings of togetherness with other people.
What 2L brought to the table was the virtual environment. While many people find that online gaming brings a strong feeling of togetherness, I would argue that this feeling is driven largely by the common purpose present in most games. It doesn't matter whether the game is an MMORPG, or a shooter like Battlefield 3. The game provides the purpose, and the gaming network provides the social conduit. This doesn't mean you can't have deep and meaningful relationships through these networks; just that the game provides the impetus.
With Facebook and Twitter, the common purpose exists in the real world, which makes the virtual reality of 2L redundant. Most people simply don't need it to feel connected while watching a TV show, finishing their run, lounging around reading online, or attending a concert.
However, i think gamers still experience it on more "regular" MMORPGs. I know I used to.