Well, not quite the push of a button - but pretty damn close.
Beforehand, setting up my Elgato Game Capture HD with a console was a wee bit of a chore. I'd have to manually unplug the composite cables from my tower of video and power and plug them into the Elgato's female jacks. This was kinda sucky, but I'd also split the Elgato's cables so that one set went to an AV input of my TV along with the consoles - so I'd have to unplug and replug cables on the TV's end too.
While this doesn't take a stupid amount of time, it was plainly an inconvenience so I'm happy to finally be done away with it. I could try to explain how I fixed it to be better, but instead, here's a hastily drawn 'wiring' diagram. Enjoy!
There are two switch boxes because I'm utilising both AV1 and AV2 ports on my TV. This is also why I can't simply split the signal somewhere along the line, and have to use an additional switch box.
Now, though - instead of swapping around a few sets of cables, I simply just have to push the button for which console I want on my tower of video and power, and then select which switch box that console is connected to on the Elgato switch box. Too easy!
Talking of my tower of video and power, here's an updated photo:
Nothing really has changed there, but I replaced the AV2 box to be the same as the rest. Beforehand, I had a crappy blue one where the output plug was on the front but all the inputs out the back, which meant the output cable to the TV (and now the Elgato switch box) was all up in the grill of the rest (which you can see here). Now, all cables are nice and tidy out the back and that makes me feel really good.