Newark usually has some specials on hook up wire and sells 100 foot spools which are a good size for a project like this. You can of course usually do better from surplus/discounter outlet type places, and always check ebay (which is where I got most of my stock), but you'll tend to find lots of weird striping patterns and such - not much in the way of primary colors.
22 stranded is indeed about the smallest that will reliably hold onto a "red" crimp quick connect. No need for discrete grounds; just daisy chain 'em, perhaps with one chain per player to keep things tidy and modular. That's generally what commercially built machines do, often along with a connector for each player's wiring to make service easier. You can use zip ties or similar to bundle up your wires and keep things neat.
If you must use smaller wire, fold it back over itself once or twice then crimp onto that. It gives the terminal more to "bite" onto. It'll still be a bit flimsy from a handling point of view, but the terminal might at least stay on. I'd still avoid solid wire for this application entirely. It doesn't crimp well, and it tends to break when flexed during service or when moving the machine.