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It strikes me as a completely reasonable hypothesis that effect on income is based on the behavior students learn to adopt in a classroom rather than on what is learned. This fits well with how the value of a college degree is often less about the specific material which is learned and quickly forgotten, and more about signaling that you have the capacity to be productive and do the tasks you are assigned over multiple years to achieve a goal.
The classroom is a much better proxy for a future work environment than life at home... you have peers/co-workers being given tasks by an authority figure, and developing the skills for navigating that political environment successfully is likely to transfer later in life. If you have a teacher or classroom environment that can successfully teach you how to ‘play the game’ (part of which will be learning how to get good test scores even if the knowledge isn’t retained), you will probably grow up better able to convince your future employer to hire you, or to convince them that you are valuable and worth paying more.