So I've been doing a lot of reading on bitcoin privacy a lot lately, and I've got a question. I'm not sure if this is the right board for this, but I feel it is.

So let's assume an address has been made public and everybody knows who that address belongs to, can the owner decide to change that address and move his coins to another address without it showing on the bitcoin blockchain?
I mean can the wallet simply change the address at the request of the owner with verification like the seed phase, of course? Then on the blockchain the new address will not show where the bitcoin came from and the old wallet will just show $0 balance without showing where the coins where sent to?