That was roughly the situation in the U.S. long ago (at least for lower levels of education), but it was built on the how the culture of the time effectively restricted certain social classes (women) to certain job sets (education), which led to a relatively high number of smart, capable and at least somewhat idealistic applicants for relatively low cost (essentially by forcing greed out of the picture).
This was not, however, a bound relationship, so as the culture changed and employment opportunities broadened, the pool of quality applicants spread out over other jobs, and the educational system didn't adapt to find new ways to draw people in.
The capitalist approach in general is probably the result of someone looking at the above issue and, well, grasping at straws for some way to change things.
I've worked in a different field where my peers were mostly smart (90th percentile plus, we checked), capable (regularly tested), idealistic (audited) and non-greedy people. When polled for why they were working there, no one mentioned money. But when presented with the idea of working without pay, most countered that the requirement of having to pay the bills would force them to work elsewhere. "Greed" can be a relative term, and in the strictest sense you'd probably only find non-greedy people among those who don't have to deal with paying the bills.
Another aspect to all this though, is that even if you take money out of the picture, you're still changing this from one form of capitalism (money based) to another (capable people). After all, capitalism is fundamentally about leveraging resources. By de facto default these days, that resource is assumed to be money, but that's not always the best fit.