Here is my intuition since I don't want to write LaTeX on my phone.
The square root of the trace of the covariance matrix represents the dispersion of the data with respect to the mean.
The average pairwise euclidean distance represents the dispersion of the data with respect to the other data points.
In general, you would expect these to be similar, as the other data points are, on average, located very close to the mean.
Consider: For a set of points, the average of the squared distances from points to all other points (including the distance of each point to itself) is equal to twice the variance of the points. See the much more eloquent answer from @whuber here. So, for a set of points, the average squared pairwise distance and the average squared distance to the mean are proportional.
However, your method of calculating pairwise distances differs in the following ways:
When calculating pairwise distances, you don't include the distance of each point to itself (ie. 0) and you don't include repeats (the distance between a pair of points is only counted once). As the number of points approaches infinity, this quantity actually approaches the variance.
When calculating the pairwise distances in multiple dimensions, you are not using squared distance, but euclidean distance. Therefore, your result is the average of the square roots of the squared distances. In contrast, the square root of the trace of the covariance matrix is the square root of the sum of squared distances. In the first case, the square root is inside the summation, and in the other, it is outside of the summation.
In short, I believe the two measures of spatial dispersion are measuring very similar things, and it is only for the two reasons listed above that you are not getting exactly proportionate results.