For the USA, I’m fairly sure it’s more the idea that it’s not the government’s concern where people live. Why would free citizens have to tell the government where they live?
For the UK, it might be more the UK mantra “if it ain’t completely and utterly broke, don’t fix it”. They tend to love traditions, even if they have become somewhat impractical.
For the UK, keeping a central register also was a bit easier than one would think, as women and most men didn’t have the right to vote until the middle/late 19th century. https://en.wikipedia.org/wiki/Reform_Act_1867:
“Before the Act, only one million of the seven million adult men in England and Wales could vote; the Act immediately doubled that number. Moreover, by the end of 1868 all male heads of household were enfranchised as a result of the end of compounding of rents.”
(If I understand it correctly, “compounding of rents” is where landlords pay the property taxes that renters have to pay, increasing rent accordingly. Because only those paying property tax were eligible to vote, a side effect was that many renters weren’t allowed to vote)
