That is part of it, but as the orbit shrinks, relativistic effects start to become the dominant source of orbit decay. The most striking demonstration of this

That is part of it, but as the orbit shrinks, relativistic effects start to become the dominant source of orbit decay. The most striking demonstration of this effect is what happens around the photon sphere (3/2 the radius of the event horizon of a non-rotating black hole. The orbital velocity at this radius is the speed of light. Consider the two body problem, of a single mass interacting gravitationally with the black hole. If this mass crosses the photon sphere from the outside, it will inevitably spiral into the event horizon. If the mass crosses the photon sphere from the inside, it will either escape to infinity proceed to cross it again from the outside (and subsequently spiral into the event horizon). The more general phenomena is orbital decay that happens in any 2 body problem under general relativity. From a conservation of energy standpoint, the massive amounts of energy contained within gravitational waves come from the release of gravitational potential energy as the two orbiting objects fall towards each other.