Best part is THIS is in the “frequently bought together”
The best RCA switch I know of is this 8 input 2 output model. It does both component and composite. You could always run the output from one switch into the input on another switch to get more total inputs. With 2 of them you could have 15 systems hooked up.
Now here is one neat trick you can do: you can use an RCA Y adapter with 2 female ends and 1 male end to combine multiple systems into a single input as long as they aren't both on at the same time. I used to do this with my SNES and N64, but it doesn't work with PS2 as it produces signal noise unless you turn it off with the master power switch on the back. You do have a chance of adding noise to your signal by doing this though so don't cheap out on cables and adapters if you do it.
For the big AC bricks you can get 1 foot long extension cables or short Y cables or 1 to 4 splitter cables to move them away from the power strip.
Now if you want something cleaner I don't think having 12 systems hooked up all at once is ever going to be clean unless you have a hell of an entertainment center type thing to put them in/on. You mentioned having a Kallax and that has no place to hide cables. I have a Besta and I hide the cables and power supplies down behind the doors. I also don't leave everything hooked up at the same time. I have it setup so my consoles are displayed on a shelf on one wall and then if I want to play one of them I can easily grab it and hook it up in less than 2 minutes. One nice thing about the Besta is that it is a modular system so you can start with the TV stand I linked (or a bigger one or a smaller one) and then add to it over time.