In that case, good teachers will avoid such students, and bad teachers will seek them out. This is actually quite a highly desirable result - we won't waste a g

In that case, good teachers will avoid such students, and bad teachers will seek them out. This is actually quite a highly desirable result - we won't waste a good teacher on hopeless students, and a bad teacher can't harm them much. That might be the case in a world where there was perfect information, no barriers to entry, and no significant asymmetries. In the real world though, a brilliant young teacher may get her start working with children damaged by, say, lead poisoning, and get no results. As a result, she'll be branded as incompetent. Since she's been branded as a bad teacher, she won't be able to get jobs in high performing schools with better students and will never get an opportunity to demonstrate her brilliance. Or she might be so discouraged by her failure that she leaves the profess altogether. You can imagine the equivalent problem faced by teachers of well-off children.