The 'Streaming PC' isn't doing anything but record the video output from the 'Gaming PC', so it can devote all of its system resources to encode it at higher quality; the 'Gaming PC' doesn't have any extra strain while it's running the game at the highest possible quality.
Using a secondary PC also means I can record games that either become unstable when recorded locally (Fallout 3, if you happened to be around for that disaster of a stream), or are difficult to hook into with a recording program in the first place (Diablo 1 was a game I experienced this with).
Lastly, a PC dedicated to recording video also allows me to render videos more frequently, which should help my upload schedule remain more consistent.
I hope to have this PC ready for the end of my current game (Legend of Heroes: Trails in the Sky FC). Thanks again to everybody!