Why is the standard error used (instead of the standard deviation) when reporting the results of measurements made with scientific instruments? One explanation that I was given is that you are considering the measurements you've made to be a sample from an infinite population of measurements that could be taken with that instrument under the conditions of interest. This strikes me as kind of silly. I feel as though it would be more appropriate to characterize the measurements you've taken to be the population of interest, as opposed to modeling it as a sample from some infinite population, and that one would rather have the dispersion data on the actual measurements that were taken. I don't see, practically speaking, how one could conclude that you would get a value closer to the standard error if you had taken a bunch more measurements.
 A: In this case the report presumably states the mean of the measurements and the "standard error of the mean".  
It is reporting the scale of the uncertainty associated with the reported mean, not the dispersion of the individual measurements around that mean.
A: The standard error of the mean is a standard deviation. Specifically, it is the standard deviation of the sample mean, regarded as a random variable of its own right. That is, it measures the spread (via a standard deviation) of the sample mean if many means were taken across across many different samples. We give it a different name because we already have something called a standard deviation, and it refers to the spread of a single observation instead of the spread of an average of observations.

I will confess that I'm not sure what you mean here:

I don't see, practically speaking, how one could conclude that you would get a value closer to the standard error if you had taken a bunch more measurements.

You wouldn't expect to get "a value closer to the standard error with more measurements". Perhaps what you're thinking of is: with more measurements (specifically in the form of a larger sample size), the standard error would tend to 0. But this should make sense; intuitively, it should be reasonable that with a very large the sample size, your sample mean should be quite close to the underlying true population mean. A standard deviation doesn't do this because it's measuring something else entirely.
One thing that might help ease your objection about choosing one over the other is recalling that in common situations (i.e. large populations and reasonably large sample sizes), they are tied together by a straightforward relationship: $\textrm {SE} = \textrm { SD }  / \sqrt n$. From that perspective, it really can't make sense to object strongly to one in favor of the other, as they're morally equivalent.
