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 :-).