I think you would be fine with #14 bus wiring as shown in Volphin's diagram above and #16 or #18 for your drops to the track. As I understand the KW's (I am not real familiar with them) they are 190 watt for 2 trains so I would guess you would be using half of the one for each larger oval and passing siding, or about 5-6 amps or so for each loop, and the other about the same only with some going to accessories.
The OGR wire is very nice, I really like it and use it on my layout, but if you use the suitcase connectors shown in G3750's post it could be difficult to remove the outer jacket to get to the wires you need to splice into for the track drops. The terminal strips would be better suited for the OGR wire. Single conductor wire (1 each, red & black) would probably be better for use with the suitcase connectors.
I personally like the euro style terminal strips, that's what I use on my layout. Here are some 10 amp euro style terminal strips on ebay, similar to what I have. Got most of mine at Radio Shack a couple years ago, they were closing out that particular model and they were pretty cheap. RS still has them but a different brand or something
I am not at all familiar with the American Flyer 3 rail switches either. I would imagine the thermostat wire would still work for them, however it is solid wire. I don't know what kind of terminals are on those switches? I am using terminal strips on most everything but my switches, they are wired directly.
I am using the thermostat wire for accessories and lighting (just about everything but track power) on my layout. I started out to use #18 or #20 stranded wire in 2, 3 and maybe 4 conductor (my preferred type), similar to the OGR wire, but it was difficult to find and most of it was shielded and more expensive than the t'stat wire. The t'stat wire was also readily available for me locally at Lowes & Home Depot. No shipping and handy, I can just go get more if I need it.
