Ok, I stand corrected. And I thought I had a decent idea too... *sigh*
Hey, no worries. I wasn't trying to organise a pedantry contest - it's funny how the mind works sometimes!
Your video is very interesting and similar to what I had in my head already, so that checks out.
Interestingly in the video I posted I noticed data said that the Enterprise is generating 12.75 billion gigawatts (or 12.75 exawatts). This really is a ridiculously big number!
For reference, it's several orders of magnitude more energy than all the energy from solar radiation hitting the Earth's atmosphere. The total power used by humans worldwide is somewhere in the order of 10 terawatts, which is one million times less than this number.
What are they doing with all that energy?
Incidentally, if we plug 12.75 billion gigawatts in to E = mc^2, assuming perfect annihilation of matter and antimatter, the Enterprise-D is consuming 140kg of fuel per second, 70kg of matter, and 70kg of antimatter. Not bad!
He said "12.75 billion gigawatts per...." and then was cut off. Could have been per second, per minute, per hour, we have no way of knowing.
What are they doing with all that energy?
Bending space-time to travel faster than light and powering weapons that can melt through continental plates I'd guess.
We could imagine that the future warp drive might somehow be an extension of the hypothetical Alcubierre drive, which itself is theorized to require far more energy than what you described to propel a ship across the galaxy. Even if they found a way to make it work with a fraction of the energy it would still require an absurd amount.