The capacitor would likely make less difference at full throttle than low throttle as the peak current draw is higher at low throttle, just a lot lot shorter. This is why multimeters don't tell you what is going on.
There are 2 things that FETs need to handle peak current, usually at low rpm and total power at high rpm, less current but for longer. this is why you always use an esc with a rated current higher than you need or measure and they often quote peak current higher than flc, this is not for brief overload but to cope with low rpm current peaks.
You can measure current at the input because the caps smooth it out, but at the output you are only measuring mean voltage across 2/3 of the windings at any one time.
As long as your not overloading the motor / esc its likely an esc problem, for what a 50A esc costs from hobbyking id just replace it maybe go to a 60A
What size prop do you have on it?
This shows what is happening and explains it
http://forums.offshoreelectrics.com/...less-Motor-ESC
Have a read of this