I was under the impression that the mean elements are given by taking the average over an orbital period of the satellite.
That is the wrong impression. To inject some somewhat crude humor, "It depends on what the meaning of the word 'means' means."
The Brouwer-Lyddane short transformation is a transformation between osculating elements and Brouwer-Lyddane short mean elements such that variations in the mean elements due to a non-spherical body are minimized over the short term, e.g., a few days. Over longer spans of time, the Korai mechanism might kick in, resulting in changes in the Brouwer-Lyddane short mean eccentricity. I'll ignore that as you're specifically asking about short mean elements.
For simplicity, suppose a vehicle is orbiting a body that has a non-zero J2 term and all other higher order gravity terms are zero. The vehicle's state is initialized using Brouwer-Lyddane short mean elements with an extremely small (or even zero) mean eccentricity. Suppose the vehicle's Cartesian state is propagated using a high quality integrator, and the osculating and mean elements are calculated from the propagated state at regular intervals. Over a short span of time (e.g. a few days) those mean elements will barely budge (except of course for mean mean anomaly).
In particular, the mean eccentricity will remain extremely close to zero. The osculating eccentricity cannot average to zero because eccentricity is always non-negative. At extremely low mean eccentricities the mean eccentricity will inevitably be well removed from the osculating eccentricity. That is exactly what you're seeing in your first plot. As the mean eccentricity increases it does indeed begin to look like the more standard meaning of "mean" (i.e., average). That is what you're seeing in your second plot.
But that is not what "mean" means in the context of mean elements. What "mean" means in this context is that the mean elements are somewhat analogous to the corresponding osculating elements but are transformed in a manner that reduces the variations seen in the osculating elements to a minimum.