If we apply this to the Cauchy distribution, your sum is one of the Riemann sums on the way to (aside from a constant factor) the integral from -pi/2 to +pi/2 of tan x dx. This integral diverges because at each endpoint it's like the integral of 1/x, but your procedure is a bit like a Cauchy principal value -- it's like taking the limit of the integral from (-pi/2+epsilon) to (pi/2-epsilon).
So it seems like it might misbehave interestingly for distributions with oddly asymmetrical tails, or with singular behaviour "inside", though "misbehave" is a rather unfair term (you can't really expect it to do well when the mean doesn't exist).
I'm not sure how we could answer question 2; what counts as "effective"? Perhaps an extension of the notion of mean is "effective" if it has nice algebraic properties; e.g., pseudomean(X)+pseudomean(Y) = pseudomean(X+Y) whenever any two of the pseudomeans exist, etc. I suspect that that isn't the case, but I'm not sure why :-).
but your procedure is a bit like a Cauchy principal value
Interestingly, we can imagine doing the integral of G (the inverse of the CDF) that you define. The Cauchy principal value is like integrating G between x- and x+ such that G(x-)=-y and G(x+)=y, and letting y go to infinity. The averaging I described is like integrating G between x and 1-x and letting x tend to zero.
In a previous post, I looked at some of the properties of using the median rather than the mean.
Inspired by Househalter's comment, it seems we might be able to take a compromise between median and mean. It seems to me that simply taking the mean of the lower quartile, median, and upper quartile would also have the nice features I described, and would likely be closer to the mean.
Furthermore, there's no reason to stop there. We can take the mean of the n-1 n-quantiles.
Two questions:
Note the unlike the median approach, for large enough n, this maximiser will pay Pascal's mugger.