I realize this is an old post and hopefully got figured out, but to help others who may have similar questions;
The "tinyESC" motor controllers have a BEC version and a non-BEC version. BEC of course means Battery-Eliminator Circuit (it has a 5V regulator that supplies the receiver with power instead of needing an additional 5V battery for the receiver). Your non-BEC version will not supply power to the receiver and hence appears to not work until you plug in the BEC version as well.
Plugging in the BEC version powers everything up until you plug in the servo. The reason for this is that the voltage regulator in the tinyESCwBEC can only supply up to 100mA and a servo tries to draw much more than this. As soon as the servo is plugged in the electronics are browned-out (barely enough power to turn on) and things stop working.
To solve this you will want to get a larger (more available current) BEC. My favourites are from Dimension Engineering. There are several versions that can supply current to many servos at once.
Additionally, the two different tinyESC/tinyESCwBEC have been upgraded and merged into tinyESC v2. This single version has a BEC in every controller, at the price of the old non-BEC version. Stores are changing over as they sell out of the v1 controllers so you will see them soon. (Or go here: http://www.fingertechrobotics.com/products.php?cat=Motor+Controllers
Hope that helps!