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?
No there is no way that you can make a transaction without registering it on Blockchain and that's the beauty, every transaction is recorded on blockchain and one with basic knowledge can easily know this wallet send funds to which wallet but if you want to remain anonymous and don't want anyone to know to which wallet you are sending funds then you can use any obfuscating tool which will use coinjoin to mix your coins and the source will be hidden.
This is the only way to do this IMO.