That's the key part. We actually have good reasons (conservation of momentum) to believe that the revolution period does not change significantly over that timescale. There's nothing that should significantly change the momentum of the orbit around the Sun by that amount/time.
Contrast that to the Earth/Moon's rotational period, which we expect to slow over time due to energy consumed by tides "sloshing around".
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.
Contrast that to the Earth/Moon's rotational period, which we expect to slow over time due to energy consumed by tides "sloshing around".