Resolution of problem posed
Many thanks to Dr. Paul Watts for helping me out with this.
The key to this problem was, as I suspected, that I was misinterpretting the chevrons.
Specifically, this assertion is wrong:
$$e\langle\cos\theta\rangle=0$$
Here, I saw that we were averaging a cosine function over a whole period, and took it to therefore be zero. What I failed to consider, however, was that the argument does not scale linearly with time, why should it! (Unless it were a circle). If one writes the argument in terms of time, the average gives the expected answer.