Note that the U.S. already buys this argument when it comes to primary and secondary education. It only seems to mostly ignore it when it comes to the university or trade school level.
The U.S. gets by without public higher education, but that might be in significant part for two reasons: accumulated private wealth in a large middle class that got rich from globalization with one of the few intact industrial bases in the post-war environment (now dwindling), immigration of educated workers from all over the world (brain drain from other countries).
Also, note that many countries with public higher education see it as a system where each individual "pays it forward" for the education of the next generation, at a moment in their lives where they have significant more economic safety than they would in their late teens. Simply put: "I'll pay for the education of the next generation, because the previous generation paid for mine". This is something which the U.S. does within families anyways, but some other countries do collectively, resulting, again, in compounded gains from a more educated society and cheaper education costs due to collective bargaining. Things like fancy dorms and millionaire school administrators are hard to justify when they are paid out of the paystub of your average nurse and construction worker...
Don't get me wrong, I did my undergrad in a (developing) country with public higher education and my graduate studies in the U.S. (on school support). There are definitely benefits to the U.S. system if you have the money or are in the top percentile for talent/luck. But overall, for the majority, the lack of a strong free public education option is a huge disadvantage, it seems to me, even after six years here, that this is one of those things where the U.S. does well despite of it, rather than because of it. Like private healthcare and lack of public transportation.
Does all tertiary education should be "free" for everyone? English history, art, etc? How does the price setting works: government just pays whatever educational institution asks for? Does it forces institution to accept the amount of money being provided?
Also, what does "better payoff for the economy" mean? Does it mean that the median wage rises, amount of taxes collected increases, productivity increases, something else... ?
As to why one would want to do it, economic activity would be higher. Lets compare two cases:
One a student goes to college, ends up with $40k worth of loans, starts working a job which they start to pay back almost immediately. The loan starts to reel back in the cost of the capital almost immediately, and it offsets the net income of the student. They spend less than they would have, they save less, and have less time to compound interest on smaller savings (creating a higher social burden). In the early portion of life you have a lot of economic activity. Moving into a first apartment, then first house generates all sorts of ancillary economic activity. All of this is delayed by higher student debt, and reflected in data.
On the other path, the same student goes to college, gets the same job at the same gross income. Now though, they have more disposable income and savings. More disposable income, means more spending, which means more economic activity. The gov't, even if it goes into debt itself to fund the education, wins in the short term with more activity and more tax receipts, and in the long term - better savings translating to a lighter load on backend gov't services. Students and parents benefit, the economy benefits, the govt doubly benefits, society improves...