No, to be more accurate, it shifted the cost onto many other people. And that's an important distinction, because when you divide a large number by a large number, you get a small number.

Think of it with this simplified example, using numbers that are easier to grok. Let's say I've got a school where tuition costs $1,000. Alice cannot afford that on her own. But there are 1,000 people in her country, so if taxes paid for that tuition, each person only needs to pay $1 extra in taxes for Alice to get a quality education. Of course, it's not just Alice going to college, but it's also not every person in the country. In fact, at any given time, most taxpaying people in a country are not attending university, since university is usually only 4-6 years and most people live to age 78 or higher.

So to adjust the example to account for that, let's just assume every member of this society of 1,000 people attended or will attend university (an overestimate for sure). Using the ages as a guideline for proportions, if 60% are above age 18, and at most 8% are currently in college (6 years at college / 78 years of life), then we can easily see how much each taxpayer must pay for everyone to get an education: (80 people * $1,000) / 600 taxpayers = $133-ish.

So you tell me which is a better society: one in which everyone gets a good education for about $130, or one in which some people can't afford a good education because it costs about 7.5x as much money?