>> If we stopped making it a game, without points to earn and rewards for points, people wouldn’t cheat.
> Nonsense.
> There is an incentive to cheat any time there is a test, where capabilities are measured that are not routinely used.
So stop issuing tests.
Or more importantly, the schools could stop holding a record of students grades. Students could still takes tests to confirm their understanding, but their results wouldn't benefit (or hinder) them in any sort of hiring situation. If you sit through the classes, assuming 1) you've put in the work 2) the classes were taught well and 3) you posses sufficient intelligence/aptitude, you should finish with some useful skills. However, no one "graduates" in the sense that they currently do: no fancy paper slip to mount on the wall, no gold stars. School then has nothing to do with career outcome, at least not directly.
This might seem unreasonable -- the employers of the world need some way to judge their fresh candidates!
This is easily solved: people new to a field would start out as apprentices. With a year or more of experience on their resume, they may now switch employers as readily as anyone else does now -- employers, in my experience, don't often check their candidates' formal education at that point.
But what about people who, at that point, would manipulate their way into a rewarding career without yielding any real value in return? At that point, we've just shifted the "game" to "get paid while being lazy and/or useless", right?
Well, maybe, but nothing has really changed. People already brown nose. Companies are already rife with cronyism/nepotism/etc. Lazy and incompetent employees already exploit past association with big name companies to coast through their career and into cushy retirement. None of that has changed.
The only things that really change in this hypothetical: universities are no longer the gatekeeper to people's future, people no longer feel the need to take on massive debt to study disciplines that they're not even sure they'd enjoy, and employers would be expected to invest a tiny amount (modest pay; the overhead of letting them shadow established employees) in new hires making their first entrance into the working world.