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The more "adjustments" you do based on demographics, the more likely that one of your adjustments is incorrect and obscures problems or encourages incorrect solutions.

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This is an argument to extremes, and it falls apart when you go to the other extreme: if you refuse to make any adjustments then you're now unable to separate the marginal effect of the school from the much larger effect of socioeconomic status. This would likewise obscure significant problems and encourage incorrect solutions.

The correct approach is to adjust for demographics but do so carefully, rigorously, and consistently.

Controlling for demographics is done all of the time in social and behavioral sciences.

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