I understand that gravitational force is a result of the curvature of spacetime and that the analogy with electromagnetic waves falls apart at a certain point. However, I have been confused when reading about gravitational waves as it is unclear to me whether they are causing the gravitational force that matter exerts on spacetime or whether they are just a byproduct of that force in certain situations.
Classical gravitational waves (aka gravitational radiation) do not cause the gravitational force. There are many situations where gravity is significant but there is no gravitational radiation. For example, consider a spherically symmetric star or planet. There is some internal pressure pushing the object apart, and the gravitational force holds the body together. The system is time independent and spherically symmetric, so there is no radiation. Much of the intuition from electrostatics carries over here.
I suspect the reason why you are confused about whether gravitational radiation is the cause of the gravitational force is because this language is often used in the context of virtual particles for the full quantum theory. Gauge bosons (like the photon or graviton) are force carriers, and virtual exchange of these generates a classical force between objects.
Clearly to fully understand the answer to this question, you should read up on gravitational waves and also QFT, but the short answer is that virtual photons/gravitons can be seen to be responsible for the generation of force