I'm not sure about this analogy. You can argue that the apparent motion of the wave crests is an illusion being pieced together by our brains when we see the totality of the elliptical movements of water particles at the surface.

But at the moment when you drop a pebble into a pond, there are definitely parts of the surface which are moving and parts which are not, and the influence of the energy you introduced with the pebble can clearly be seen to spread outward over time.

Granted this doesn't map directly onto electromagnetic waves because the mechanisms involved in wave propagation are different.

This picture is incorrect: the electromagnetic wave has a mechanical momentum in the direction of its propagation, which means that something is moving in that direction.

Does it have momentum before we measure it? I thought momentum was a property of the collapse event, not a property of the wave?

According to the classical electrodynamics - it sure does. From the quantum mechanical point of view, it also does - in the sense that we can always measure it (i.e. it is an observable). The "property" in this case is not so much a particular outcome of such measurement as much as the expectation value; actually, I'm afraid that the use of the word "property" in this context can only lead to confusion as it effectively conflates several different things: the (quantum-mechanical) state, the observable, and the particular value observed.