> There's usually much more variation between individuals within a group (say, all white people) than between groups (say, white vs. asian people), nevermind individuals from different groups.

This is statistically illiterate. Let's say you have two sets of random variables X and Y, both from Normal distributions with standard deviation 2, where the mean of X is 0 and the mean of Y is 1. Knowing whether a measurement comes from X or Y will still allow you to make more accurate predictions, even though the within-group variance is larger than the between-group variance. For very large groups, this applies much more so. If you have high-dimensional multivariate data, it is possible to assign individuals to clusters very accurately even if all individual measurements overlap substantially.

See for example http://www.ncbi.nlm.nih.gov/pubmed/12879450

> Not to mention morally dubious, at best.

This is the real issue: the use of certain stereotypes is a political and ethical debate, and those against using stereotypes should stop pretending there are no costs in terms of sub-optimal decision making. There are defensible ethical reasons for being against some types of discrimination based on valid stereotypes.