I wonder why a commercial entity that registers ownership / titles for free, and bills for checking, did not spring up. Clearly there is moneyed demand for certainty about title rights, and if you can provide certainty (because the last deal was registered with you), it may be a more desirable product than mere insurance.
That is more or less how title insurance started back in the day.
In 1800 land was sold in person only by people who knew each other, in front of other witnesses who knew everybody in town. It worked great, which is why some states (I assume like CT) never bothered with a registry. In the mid 1800s as land out west started opening up for settlement (skip the whole bad treatment of the natives) investors "out east" wanted to invest in land and ran into a problem: they didn't want to go out to the land, but they knew scams existed so they started hiring trusted people to travel instead and verify they property owner was really the person they were buying from. Some states have a registry and so you don't need that, the state tracks owners and verifies the people buying/selling really are who they say they are.
It’s turtles all the way down, though — how do you know that last deal had clear title? At some point you’re going to want insurance anyway.
In any case, the US system is already that the government records ownership (not for free, but for a small recording fee) and the title company charges for checking, and for insurance in case they get it wrong.
As just one example of how it can go wrong, here in Seattle it’s common to find out your lot is nine inches smaller than you thought because surveying technology is a lot better now than it was when your deed was written in 1908.