Exactly, profiling.
You can’t say if [some race] then that. But you can use other things (are they loitering, barefoot, poor hygiene, has sunglasses, truckers cap, pulls over hoodie on entering, have a decoy, etc)
If the proportion of white shoplifters and black shoplifters being caught is different, AI will just scale up that disparity. If reality is biased, we should work to fix that: not make it more efficient.
To specifically address your point:
"so long as their accuracy is within reasonable range of effectiveness"
This is a far too narrow look at what accuracy means - that's why we usually look at sensitivity, specificity, false negative rate, and false positive rates, and why we break those rates up over important subgroups. To the example numbers you give, let's posit (without evidence) that your numbers are correct - they don't tell us what proportion of people visiting the store belong to each group (is the model just learning to identify ethnicity?), and they don't tell us what proportion of each group is really shoplifting.
In many areas (drug crimes come to mind), there's good evidence that different ethnic groups commit crimes at similar rates, but arrest rates are wildly, grotesquely disproportionate. Imagine that AI is applied to such a system - it learns, correctly, that black people are more likely to be arrested for particular crimes. It then targets black people, and the accuracy numbers look good on paper. In fact, because of the trust we have in technology, such a system might help normalize or justify this crazy preexisting disparity.
If you were building a statistical model to describe predictors of shoplifting, this kind of bias wouldn't make it past a cursory level of peer review. When you're slapping "AI" on a product and implementing it at a huge scale, nobody bats an eye.
My point is about the data generating mechanism - making inferences conditional on a bad selection procedure is a bad idea. It would be bad statistics, and it's bad ML. In this case, my argument is that it's also bad for society.
For example I can legally profile short people, but it isn't moral to do so. And I would rightfully be judged if I did so.
Asking if it is right to judge people based on their mental health is an entirely different topic than asking if we can judge people based on a physical characteristic they were born with and have no control over. Apples and oranges.