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One way to have skin in the game is to pay a considerable sum of money to join medical school. This reminds me of the Commitment and Consistency chapter in Cialdini's book Influence. In part it says we stick with things more after we exert effort and especially so after public effort. With free tuition we take away some of the skin in the game, but students will still have to expend considerable effort to achieve high enough grades. That probably fosters commitment (and might explain why people with scholarships still tend to value school). In any case, we already have doctors whose parents paid for school, so even when free schooling does reduce commitment (and it probably will sometimes), at least this evens the playing field.