Testing knowledge or intelligence amounts to measuring aptitude and potential. Aptitude differs from knowledge by reference to a virtue, a culturally defined norm.

Applying knowledge and intelligence tests to an individual, a cohort or a population amounts to:

a: measuring the aptitude or potential of an individual,
b: the average aptitude or potential of a cohort,
c: the performance of a teacher in creating aptitude given a cohort,
d: the average aptitude or potential of a specific population,
e: the performance of an institution in creating aptitude given a specific population, or
f: the performance of an institution in creating aptitude given the mean of a general population.

By design, a knowledge tests (i.e. command of elements of the curriculum), not only measures technical proficiency but also marginal potential of an individual given a standardized curriculum (cases "a" and "b").

If intelligence is considered a potential that depends, predominantly, or fully, on the individual then measuring intelligence in a specific population, with the implicit intention of comparing it to other populations, amounts to testing a hypothesis regarding the distribution of phenotypes, i.e. statistically significant genetic differences between populations (case "d").

This approach is likely to lead us back to the Social-Darwinist theories of a racist nature of the late 19th to mid-20th century. In a Darwinian framework, an educational institution, or an individual teacher, can't be held responsible for sub-par performance of a cohort.

This changes in a Lamarckian paradigm: if the socio-cultural context, e.g., attitude to leaning in infancy, preschool training, among peers, etc, is part of the culture of an population, then aptitude tests of a cohort or a population would measure the performance of the socio-cultural system with respect to "creating potential" [1].

Here lies a problem that's best demonstrable when questions regarding common-sense [2] (i.e. culturally normal knowledge) are applied as a proxy for measuring intelligence: the measures of aptitude and intelligence then are covariate.

It would, of course, be interesting to apply a paradigm where case "a", optimal training of an individual, or a Lamarckian evolutionary approach to case "d" for improving the aptitude of a population including the embedded education system (as a side note, the opposite is demonstrably true: by limiting the performance of an education system an entire population can be destined to a low average intellectual performance - the history of slavery in the 19th century provides ample anecdotal evidence).

There is a catch, however: the art and science of education is by no means different from other socio-technical domains. The desire to govern a system of interest (e.g. a school district) is thus akin to governing a business entity, and the same meta-technology, cybernetics [3], is being applied throughout institutions, e.g., by the way of memetic contagion [4][5].

By applying aptitude measurement to assessing the performance of the teacher (case "c") the governance of the cybernetic system "education embedded in a population" is bound to be maladaptive: the teacher, and as well as the institution, is likely to engage in what's known as "reward hacking" [6]. This diminishes the value of the aptitude measurement not only for the intended case "c" but also for cases "b", "d", "e" and "f"!

It would be laudable to measure the intelligence of the individual and their aptitude by applying assessments tests in a longitudinal regime. Such assessments would best distinguish between individual learning traits and knowledge or proficiency in an objective way. Doing that would open the door to tangible improvement of the performance of the individual, a cohort, the teacher, an institution, and a population as a whole.

[1] Wilkins, John S. "The appearance of Lamarckism in the evolution of culture." Darwinism and evolutionary economics (2001): 160-183.
[2] Geertz, Clifford. "Common sense as a cultural system." The Antioch Review 33.1 (1975): 5-26.
[3] Ashby, W. Ross. An introduction to cybernetics. Chapman & Hall Ltd, 1961.
[4] DiMaggio, Paul J., and Walter W. Powell. "The iron cage revisited: Institutional isomorphism and collective rationality in organizational fields." American sociological review (1983): 147-160.
[5] Marsden, Paul. "Memetics and social contagion: Two sides of the same coin." Journal of Memetics-Evolutionary Models of Information Transmission 2.2 (1998): 171-185.
[6] Slater, Avery. "The Golem and the Game of Automation." 2021 IEEE Conference on Norbert Wiener in the 21st Century (21CW). IEEE, 2021