It works like this, if you just let the market decide then a few big dairy producers will drive all of the other dairies out of business and the only milk products on the market will be of lower quality than they are today. If you prop up the market with subsidies and minimum prices you force the cost of milk to rise and natural economics causes people to substitute something else that is less expensive. But if the subsidies are paid to the farmers, and not to the consumers, the farmers keep producing milk products even when the market doesn't want them at the price they are producing them.
Places that would like dairy products outside the US are already protecting their own dairy farmers and so they are tariffing the heck out of milk (see the Canadian discussion as part of the USMCA proposal).
Many dairy products are perishable with a short life although a big chunk of my uncle's dairy output ended up as non-fat dry milk. This makes it harder to store over periods of lower demand, and cows produce at a fairly constant rate whether you like it or not. Supply management is tough.
Milk and cheese would be a good candidate for a basic income sort of scheme. Where everyone would be issued milk and cheese vouchers every month that could be exchanged for milk and cheese at the local package milk store. People who didn't want their coupons could give them to people who did. That would create a 'base layer' of milk production and consumption that would keep dairies around but take milk and cheese out of the cost of living equation. I'm sure there are other reasons why this is impractical but the whole problem is fairly intractable as it is without putting farms out of business.
EDIT: One interesting thing would be to tie the price of sugared drinks to the price of milk. So it wouldn't be cheaper to give your kids soda than milk. Again, impractical but it is a challenging problem.