Anyone that thinks it's remotely comparable doesn't understand it.
It's not that hard to understand. In terms of the actual experience of playing the game, LAN party > split-screen. All of your criticisms have nothing to do with the actual gameplay experience and everything to do with convenience.
1) That's why co-op has become so popular again.
Since when does co-op require split-screen? Welcome to the internet. Split-screen is certainly not the reason why co-op has become so popular again. Appealing to a broader, more casual audience is the reason why co-op has become popular again. Co-op is simply the least intimidating way to play games for casual players. It requires very little effort to be enjoyable because it isn't the game itself that brings the joy but the act of playing the game with a friend.
2) Most people aren't capable of focusing on all that at once. It surprises me how many people play a good amount and never learn to monitor other screens.
It doesn't take much focus at all. An occasional glance is all you need and it's all that much easier when most split-screen games only support 2 players.
3) For those that can, it's part of the game. Goldeneye's biggest flaw was no bots, meaning 3 or fewer people would get into standoffs easily. But throwing equally matched people into a game was always fun. Monitoring other screens becomes strategy. You know they'll see what you're doing so you need to find ways to either trick them or simply beat them despite their eyes.
In other words, trying to compensate for the inherent stupidity of sharing a screen in a competitive deathmatch experience. As you said yourself, standoffs were common because you could always see what your opponents were doing. Trying to pass this off as some kind of benefit is like arguing that using a gamepad for shooters makes the experience more intense and realistic because it's harder to aim.
As you mentioned in an earlier post, anything is more fun with friends. This includes playing crappy games with crappy control schemes and crappy display modes. Saying that something is fun in co-op is utterly meaningless because almost anything is fun in co-op.