So, I always thought the idea of bootstrapping was that you have a sample from which you obtain an estimator for some function of the population (like the average height). And then when you bootstrap by resampling, you get draws from the distribution of the estimator (and hence the variance). I took it for granted that the mean across the bootstrapped samples would be the same as the mean from the original sample. This is definitely true for statistics such as any average across the population.
For any nonlinear function, however, such as the percentage of "rich people" in the sample, the two means will be different. So, bootstrapping is in effect telling you that your original estimator has a different mean now (which is in most cases also the mode).
Given this bias, is it still appropriate to use bootstrapping to measure variance (you obviously won't use the bootstrapped mean in place of the original estimate)?