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.
The refusal to write blank checks is more of an exception than the rule. For at least a couple of decades, politicians have begged young people to take any university education, any at all, with many nominally highly educated graduates out of work.
The government pays a $1000/month (pre tax) stipend to all students for up to six years, and offers a similar sized amount in loans on very generous terms. As tuition is free, this is only to cover living expenses and study materials such as books. Taxes are high, even at that income level and so are living expenses, but it's still not a trivial sum.
Those aren't blank checks though. The problem with the US system is that the government is backing loans with no regulation of what the universities charge, with the result that tuition costs have increased to the point where most middle-class people couldn't afford to send their kids to school if it wasn't for loans.
Government paying education costs and regulating the price can work. Government paying education costs and not regulating the price is just corporate welfare.
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.