The government already does tell you what you can and can't do for a living in regards to classical professions. You can't just hang up a shingle and call yourself a lawyer, an engineer, or a doctor (or other healthcare professional).

The only difference with the Danish example is that the government refuses to write a blank check (student loans) to law schools who want to take advantage of the naivete of students.

The US system would be vastly improved if the government stopped subsidizing the education industry's market distortions, or if the government actually regulated the education market. Right now it is the worst of both worlds with the government helping the education industry to fleece students and other tax payers.