The board didn't work.
Or at least it didn't before.
My previous post was a false positive and everything started to behave really weird, with negative voltages and what not, as soon as something was connected and plugged in. The power switch was controlling a MOSFET, which in turn disconnected or connected all power lines to GND. For some reason the current flow was really wack because of this. For instance, inserting a LED in the 5V line really shouldn't reroute power from the 3.3V line and vice versa. This was a real pain to troubleshoot, but after butchering the board quite badly (#sadface) I ultimately found the problem (#happyface).
The solution simple enough - just bypass the MOSFET by shorting the drain and source. I really wish that I'd found this simple solution before massacring all capacitors. Now things are finally working correctly with the exception that I cannot turn power on and off with the switch, as I've hoped to do. But it's a minor issue and was more of a nice touch anyway, as it wouldn't be used in production.
|1) Everything's looking dandy! For real, this time...|
A major give-away, which I can't believe I overlooked, was that the status light previously didn't light up. It's directly connected to pin 13, so it should have blinked during boot. The fact that it never once did (before) would have told me that current was going the wrong way - or was insufficient. But I was simply assuming that I was shorting stuff when in reality it's perfectly fine to measure where and how I did. The wonders of stress...
Another issue leading to "premature optimization" was that I'm using an old Chipkit, which of course had some pins enabled that are now used for MOSFET control. This lead to having the light rows being always on for instance, but since they turn off during reset of the Chipkit they work perfectly fine.
Lesson learned? Don't put in untested parts in the schematics on the basis that "it should work, right?". The power switch was a last minute addition and a huge pain the Arizona.