What do they do with all the food that doesn’t get eaten?
Every table, a buffet of tiny servings surrounding a large rice bowl.
30+ dishes isn't uncommon to serve to a group. The structure is less a statement about food abundance (which is definitely what the article is claiming) and more about the variety historically available to the class which can afford to eat in sit-down restaurants or be fed by courtly chefs.
The traditional Chinese sit-down restaurant experience (to grossly generalize) isn't that different; There might be as many entrees as there are people, using large serving bowls, but each person is by default expected to take a fraction of each entree.
But yes, the leftover dishes are thrown away.
Many middle-aged and older customers have a habit of mixing up leftover vegetables after a meal, and they encourage others to do so. The idea is that if everyone does the same, the restaurant can't reuse any dishes.
Meanwhile, honest restaurant owners want to assure customers that the their dishes are new. So they serve food in a way that will make it very obvious if it was reused. For example, kimchi is often served in long (~30cm) slices, so that customers will have to cut them themselves, as if breaking a seal.
Nice to learn that bit about long kimchi slices. I just assumed it was because people like it long, like with spaghetti.
So you're more likely to be served uncut kimchi in restaurants where the menu requires fresh crisp kimchi, as in bossam, kalguksu, and seolleongtang.