Bootstrapping and Estimators I have a textbook that states something to the effect of:

The mean of a bootstrap distribution is not an accurate approximation
of of the mean of the sampling distribution.
But, the spread of the bootstrap distribution is an accurate
approximation of of the spread of the sampling distribution. (I'm
guessing this means standard deviation.)

Is this correct?
I realize why it's pointless to use the mean of the bootstrap, that's not what I'm asking. It seems like the standard deviation/variance are just as "inaccurate" as the mean, and the bootstrap distribution isn't any more "accurate" of an approximation of the sampling distribution than the sample mean is of the population mean.
Is there some measure of accuracy for these estimators that makes them better?
 A: This is actually quite a complicated question, and it mostly isn't specific to the bootstrap. Some of what I will say initially isn't true, but I'll follow up by explaining why it's not really misleading.
First, let's fix a concrete example.  You're interested in estimating the compressive strength of a type of concrete, and you have $n$ samples $X_1,\dots, X_n$ of concrete taken in a suitably random way that we won't go into in detail.
If $n$ is large, the mean $\bar X_n$ is close to $\mu$, the actual average compressive strength of this type of concrete. We would like to say as much as possible about $\bar X_n -\mu$, the error.
The bootstrap isn't going to tell us anything about the mean of $\bar  X_n-\mu$, as you already understand. If we had any information about the mean of $\bar X_n$ we would have used it already to make a better estimator.  But we haven't used up as much of the information in the data about the scale of $\bar X_n-\mu$.
Suppose everything was Normal and we were just interested in means. In that case the variance estimator $S^2_n$ is actually independent of $\bar X$, so (in some sense) we haven't used up any information about the variance. We can estimate the standard error by $\sqrt{S^2_n/n}$ and that tells us something about the standard deviation of $\bar X_n-\mu$. We can improve our previous "$\bar X_n$ is close to $\mu$" to a much more precise statement. For large enough $n$ we know the scale of $\bar X_n-\mu$ very accurately.
So in the non-bootstrap setting, trying to estimate the mean of $\bar X_n-\mu$ doesn't tell us anything new about $\mu$, but the standard error estimate tells us a lot about the scale of $\bar X_n-\mu$.
Exactly the same is true for the bootstrap. The mean of the bootstrap distribution doesn't tell us anything new about the mean of $\bar X_n-\mu$, because we were already using that information, but the variance of the bootstrap distribution estimates the scale of $\bar X_n-\mu$ quite accurately.
So, that's one of the things going on: we're already using the information about $\mu$ to work out $\bar X_n$, so the bootstrap doesn't tell us anything extra. There's also the issue that we care more about the accuracy of the mean than of the standard error. A standard error estimator that's just as inaccurate as our mean estimator is still pretty good.
Suppose $\mu$ for our concrete is 28 MPa, and the standard deviation is 5MPa, and $n=100$. A 10% error in $\mu$ would be 2.8 MPa -- let's suppose that's enough to care about. The standard error of $\bar X_n$ is $5/\sqrt{100}=0.5$ MPa, and a 10% error in that is 0.05 MPa, which is not such a big deal. You can phrase this in terms of maths: a given size error in scale gets divided by $\sqrt{n}$ and so matters less to inference.
Back to the things that weren't true. For $\bar X_n$ as an estimator it is actually true that there isn't any more information about $\mu$ in the data. In general, though, bootstrap bias correction is possible when your estimator is intrinsically biased; the bootstrap does give you some information about the mean of $\hat\mu-\mu$.  It's not useful very often because most of the popular estimators have much smaller bias than standard error, but there are real applications of it, such as estimating overfitting bias in regression modelling.
