Is it maybe because we are not talking about point masses, but extended objects, which makes the masses deviate from their geodesic motion?
No. Point masses can move on geodesics and make waves at the same time!
The EM analogy: We can prove, with minimal assumptions, that orbiting masses must make gravitational waves. A (macroscopic) positive charge orbiting a negative charge (or visa-versa) at low speeds is similar to gravity. An observer 1 light second away only knows where said charge is one second back in time. The electric field they feel doesn't point to the old location, it points to what would be the current location if the charge continued to move at a constant speed, see this question for more details. When the charge is accelerated (such as by another charge) the new information on it's whereabouts propagates outward at the speed of light.
Does this outward propagation carry energy? Yes! Distant objects can extract energy from the changing field (much like a float bobbing in the waves). But energy can only be transferred at the speed of light. So there must be energy (electromagnetic waves) in flight between the charges and distant observers.
The same argument applies to gravity: information that encodes the location of orbiting masses propagates out at the speed of light as gravitational waves. In both gravity and electromagnetism, this energy loss causes the orbit to decay. The finite speed of information transfer is ultimately what necessitates orbiting objects to radiate away energy.
No acceleration needed: We don't need objects to have feel a proper acceleration in order for the above argument to work. We just need objects to have an apparent acceleration from the point of view of distant observers.
In general relativity, all forms of matter, energy, and even pressure/tension distort spacetime. This means that the spacetime curvature around the orbiting masses comes from two sources: the masses themselves and the gravitational waves being created between them. The masses bend the geodesics into a circle (or a precessing ellipse, for eccentric orbits), keeping them in an orbit around each-other. The gravitational waves further modify the shape of the geodesics, causing the orbit to gradually spiral inward. The masses never feel any acceleration but they find themselves locked in a death spiral due to the curvature that they created in their own local spacetime.
Again, we have analogies to the electromagnetic world: the coulomb force keeps charges in orbit and the electromagnetic radiation emitted creates a back reaction that causes a drag force on the charges. So why isn't gravity a force, if it behaves so much like the bonafide forces in electromagnetism? With strongly curved spacetime, gravity starts to incorporate a richer set of behavior, such as time dilation and the formation of event horizons. This never happens with any other kind of interaction.