I play a ton of Nasus, so I feel you, but I still think you are wrong. Split pushing in solo queue immediately leverages responsibility onto a bunch of different parties in ways that are just destined to fail. The split pusher has to effectively push, while understand the rest of the map. Is there pressure elsewhere so if people come to stop him his team will make a favorable trade? Is he strong enough to 1v2 his lane opponent and the jungler, or at least escape? In most cases, the end result is a dead pusher and a team who isn't positioned to take an objective off it. You have to be able to react to the map to split, and you have to buy wards or have a jungler who is willing to facilitate your endeavors. On top of that, you immediately put an onus on the other four members of your team to do what is ultimately the hardest part of solo queue: not getting out of position or being baited into a fight with a numbers disadvantage. You, as the split pusher, can do your job perfectly. But more often than not someone on your team is going to get caught, and the other three are going to fight, and then you end up with a 4 for 1 and the enemy team taking two turrets and an inhibitor while you push for an inner.
I like Shen right, but this same concept is why I think he's trash in solo queue. Teams won't even wait for me to push up a sidelane near a turret before someone is caught or dead. I nearly passed out last night when I called 20 seconds on my Sej ult and a team actually waited 20 seconds before forcing a dragon fight. People just aren't thinking clearly and want to fight, fight, fight and outplay people, at least in mid gold. Nasus is different because you can draw 3 people, kill two of them, get away, or just TP into team fights and murder everyone. But playing Jax or shaco or anyone like that in solo queue strikes me as a masochistic way to play the game. Every game is going to be hugely frustrating because while you may understand how to optimally perform your role your team is probably going to do dumb shit and cry at you about it the whole game.
It's a whole lot easier to just pick an engage champion, order people around with pings, and choose when your team fights. I can't imagine helplessly trying to ping people back while you build up a wave to push with and blindly hoping they don't go full retard while you are doing so.