This is a bad idea.
Let's start off with the first point "They are just easements on private property" That's what most public roads are. They are still owned by the public (the government) and maintained by the public (the government). Where I'm from, we have plenty of dirt and gravel roads just like this. Once every 10 years or so the county comes in and re-gravels them. They are mostly only ever used by private individuals and they cross through private land, but they are ultimately public. There are actual dirt roads which are also publicly owned but unmaintained.
Where there are private roads, they've been an absolutely nightmare. What's happened with private roads is rich individuals tend to like to throw up gates and no trespassing signs on the roads. Particularly if it blocks off public land, which they don't own, and turns it into their playground. That happens all over the country. In particular you'll see it happen to lakes an on beaches all the time.
But finally, you benefit from the public road whether you want to admit it or not. And particularly when population density goes up, public roads become much MUCH more important. The model you propose is one that only works if we are in very rural locations. It completely breaks down as soon as you need to get goods into a grocery store or supermarket. We subsidize with our taxes the damage done to roads by shipping products and goods. Because I like eating fresh fruit and vegetables, I don't mind that I'm paying a little more taxes so I can have access to those goods. Along with amazon shipments and my mail.
There's a reason no ideal libertarian society exists, because they go to shit when they are tried (see Argentina who is currently looking for a bailout from the US because of their dumb economic policies that you likely support). There are plenty of failed "free town" experiments. The book "A Libertarian Walks Into a Bear" is an amusing story about exactly how this system ultimately turns into a trash heap city.