I'm writing up a couple of side points, additionally to the main ones which have been discussed in Austin Hemmelgarn's answer.
Natural salt content
Some of the salt is already there. Cows are mammals, and their body fluids contain a certain amount of salt. When you take 400 ml of milk, and make 100 g of cheese out of them (a typical proportion for a semihard cheese), you end up with 0.5 g of salt per 100 g of cheese. This is pretty much the minimum amount of salt which you're always going to have in there.
Almost all cheese also has salt added, but not all. I've actually had cheese made without extra salt, one of the larger supermarket brand organic Edamer used to be 0.5 per 100 g, but they've changed it by now to be more in the 0.7 to 1.0 range. The taste was somewhat unaccustomed, but I liked it well enough.
It may be all about taste, after all
As the other answer described, the salt has a lot of functions related to fermentation conditions, preservation and texture. It turns out that it's not the NaCl alone that can fulfill these functions, and there are, every now and again, people who attempt to make low-sodium cheese with alternative salts, such as potassium chloride. The cheese then hits all these desired points - but it also has a highly unusual taste, which most people dislike at first bite. So, strictly speaking, taste is the limiting factor which makes us use sodium chloride in cheese, specifically. The twist is that you still can't use less salt if you're OK with a less salty taste.
Young cheeses are irrelevant to the discussion
This is actually contained in the other answer if you read between the lines. Your statement that "at least the simple young cheeses can be made perfectly well without any salt" is misleading in this context, because young cheeses are different from mature cheeses in every way that's relevant to salting. They undergo a different fermentation (if any), they retain much more water, and they are highly perishable. So, they can be made without salt - but other cheeses need it.