As mentioned upthread, most people don't mind sorting on SAT but oppose sorting on IQ. Hypothesis: Any measure that is perceived (correctly or not) to measure native talent accurately will be opposed, because people are afraid that their kids may not be talented "enough", and if the measure is accurate, they won't be able to gimmick it. People want a measure they can manipulate to their child's advantage.
I don't have any evidence for or against, but it seems plausible to me. I would expect people to want a system that benefits their own kids over others; I would expect that to be something you Can't Say (because it amounts to defection); and a system that purports to be neutral but is actually gimmickable by the parent (through educational choices) would seem to suit.
[ETA: "Intelligence" as a quality is so nebulous that people won't mind that, either; if they claim their kid is intelligent, but all measures of it are rejected as invalid for one reason or another, then nothing can prove that their kid is in fact stupid. The state of being intelligent becomes a matter of opinion, not a fact. Opinions are "safe", in that they can't be proven wrong; but accepting a measure as valid means accepting that it may measure you and yours as wanting, with no "just your opinion!" or "just socially disadvantaged!" to save you]
Hypothesis: Any measure that is perceived (correctly or not) to measure native talent accurately will be opposed, because people are afraid that their kids may not be talented "enough", and if the measure is accurate, they won't be able to gimmick it.
Hypothesis: People prefer an obvious measure, not a speculative one. If you want to measure how good people are at doing something, just let them do it and measure the results. Instead of using a predictor which may or may not work.
SAT is something the child did. IQ is what other people measured a...
Related: Heuristics for Evaluating the Soundness of the Academic Mainstream, Admitting to Bias, The Ideological Turing Test