Different strokes for different folks. It depends on what you mean by "better". Different students have different desires and the US provides a wide range of possibilities. This answer applies to the US, only so far as I know.
Most public research universities are very good. Many private universities are very good. But they are different. The research possibilities are probably better, in general, at public universities, but the undergraduate classes tend to be very large with not a lot of contact with professors (more with TAs). Private universities tend to have smaller classes, maybe 30 or so, with more contact and personal attention possible.
At any university, the education is what you make of it. Yes, you can get a wonderful undergraduate education at Yale, but not everyone does. There are some very foolish US politicians who spew garbage but have degrees from Ivy League universities. It has been said (apocryphal, perhaps) that there is no one dumber than a C student from Yale. And, BTW, The Ivy League is an athletic conference.
The excellent private research universities are often old and have "deep pockets" provided in part by contributions from wealthy alumni over the many many years. They can afford to hire great faculty, and often do. And, due their age, a reputation can build on past accomplishments. But public universities have the backing of state taxpayers who, at least in the more enlightened states, value an educated populace - for many reasons.
At public research universities, in the final year or two, students have more opportunity for close contact with a research focused faculty member. This can be important for some, but not all, students.
My undergraduate years were at a very good private place that, at the time, had only about 2000 students. Classes were small. Faculty was focused on teaching. I never thought a lot about research in those years so it didn't matter to me that my professors weren't primarily researchers.
My graduate years were at good public universities where I was a TA. Classes were large and I, as a TA, handled most of the student contact, rather than the professor. Graduate classes, however, were small.
The fact that some might consider faculty at a public to be "lazy, etc." doesn't make it true. They are busy, but with research, not with student contact, for the most part.
Note, however, that the US also has a lot of smaller public universities that value teaching over research. The university systems in both New York and Minnesota, for example, have smaller teaching focused colleges, much like the typical private university. Costs are a bit less (for residents) than private places.
I live in a town with a public teaching-focused university. My neighbor told me that the expectations for faculty advancement there depended on contributions to teaching, research, and service at a 60/30/10 ratio IIRC (or close to that). Reverse the first two numbers, more or less, for a research university.
The US has thousands of colleges and universities. They range in quality from terrible to excellent.