The fuel pump ONLY has power for 2-4 seconds after you turn the key to "ON", unless the engine starts. If the engine does not start, power is removed from the fuel pump. Your thread title "Fuel pump has power but doesn't work?" is vague, you put a question mark at the end of a statement. Together, all of this leads me to believe your fuel pump is fine, but it does NOT "have power" and you are not getting anywhere because you are making invalid assumptions.

How I would troubleshoot this:
1) Strap the fuel pump relay with a rubber band and turn the key to ON.
2) Is the pump running? (You can usually hear it in the cabin, if not, remove the cover and listen (this means getting under the car).
3) If the pump is not running use a voltmeter to verify 12 volts into the pump, then measure voltage on the ground side of the pump, it better be less than .5 volt. If it is higher, fix your missing ground.
If you have power to the pump and good ground to the pump, the pump is bad. If you do not have power to the pump, replace the RCM.

Or, just replace the RCM as Zeracer told you to.

I carry a spare RCM, both because they fail so much and to help other drivers at Crossfire events. So, put your spare RCM in and see if that takes care of it - it probably will.