Update: I tried flashing just a simple blink program to a fresh ESP-12F and powered it with a 9v battery via my voltage regulator and nothing else (not on a PCB, just on my dining room table with a couple of jumpers) and same thing. Just a dim, solid, onboard LED. WTH?
That rules out that my PCB design or sketch are the problem. I don't think its my power flow either, as I've successfully used these voltage regulators and batteries successfully in the past. Every combination of pulling GPIO2, 15, RST, etc. up and down doesn't seem to have any effect. Is there something I don't know about ESP-12F's, compared to other models?