So, no, it's not obvious. Where I live that would be some combination of the county and the the town, given the Registrar of Deeds is a county position and things like property tax and liens are handled at the town level. And the county really just handles some legal matters and doesn't collect any taxes so that sort of means the state.
To quote a previous comment it's all a bit of a mess. "All" you need to do is revamp property record keeping across the US and title insurance would be less of a big deal.
The US has a lot of historical baggage that desperately needs a thorough cleanup. There should not be two (or more) "sources of truth" for things affecting a property or alternatively, notaries, lawyers or whatever y'all use to deal with transferring real estate should be mandated to warn their clients if there is more than one place where records are kept.
To quote a previous comment it's all a bit of a mess. "All" you need to do is revamp property record keeping across the US and title insurance would be less of a big deal.