There isn’t one centralized database with that information for everyone. Each state, and even inside one state each county, maintains their own records, so someone who is born in Florida, gets married in California, and then has a kid born in Oregon would need to contact each of those states for proof of those events if they lost the original copies for whatever reason. There is a national system that can (mostly) check against those state and local level records, but it also has limitations. Passports and birth certificates can be proof of citizenship, but don’t prove where you live at the time of the election even if you do have one on hand.
The voter registration process already gives the state enough information to determine whether an individual is eligible to vote in that state. For federal elections (which are still run by the states, just for federal level positions), that includes confirming that the person is a citizen, but each state is able to decide whether to allow noncitizen residents to vote on local government issues or not.
Plenty of people have tried to make that argument. Those people typically wind up spending years of their life in prison once their delinquent taxes have built up enough (and losing the property that they were trying to avoid paying taxes on, also).
Anyone who tries to tell you that they’ve successfully argued in court that the law doesn’t apply unless you sign a contract agreeing to it is either just plain lying or got lucky on a small enough case that the judge or prosecution just didn’t want to deal with the headache.
By the way, keep in mind that property rights aren’t a naturally existing phenomenon. You own your home because the state agrees that you are the rightful owner. Is a different system theoretically possible? Sure, probably, but the state-based enforcement of property ownership is the one that exists today.