I simplified. Some of it actually is consumed by friction (and therefore heating), too. You're correct that most is transferred.
However, the angular rate of the Moon's rotation is staying the same over time (it's tidally locked at one rotation per revolution). It's not exactly speeding up. Instead it's getting further away, which increases the moment of inertia and therefore transfers momentum.
That‘s what “speeding up” in a 1/r potential means, you move to a higher energy orbit, which happens to be longer and slower. Classical physics can be weird, too.
However, the angular rate of the Moon's rotation is staying the same over time (it's tidally locked at one rotation per revolution). It's not exactly speeding up. Instead it's getting further away, which increases the moment of inertia and therefore transfers momentum.