Could be a capacitor or IC shorted to ground on 3v3 output.
When you measured the 3.3V rail, resistance to ground was what?
As you said, next step can be injecting 3.3V, preferably start low and dial up the voltage, with a current limit of ~300mA. If current limit exceeded, poke around to see if anything is getting hot or if you can narrow down the short with voltage drop measurements.