Health insurance is an entirely different animal. It has its own flaws and issues, as well as its own benefits. You can't easily compare a service product and an insurance product, they're just too different.

Though yes, financially health insurance also has no monetary value when anyone can get it for free. You can't assign a price to it and anyone in the health insurance business is entirely at the whims of what the government is willing to pay them to provide a service deemed essential enough to subsidize the entire cost of the product.

Free does not mean limitless. Where I live in EU its not uncommon to wait for over a year to see a doctor on „free” insurance and less than 24h when you pay out of your pocket.

People get free insurance but hospitals get fixed amounts of cash allowing them to admit fixed amount of patients

In this scenario the answer is yes, it loses some value. Still much better system than private care in US

If everyone has a Harvard degree, the value of a Harvard degree loses value, yes.