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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 get your point, what you are trying to say is that if you have two bitcoin address and one is known by many, and you received bitcoin to wallet A and you want send your coin from wallet A to wallet B without without trace on the blockchain that will it be possible right ? well there is no transaction made in the blockchain technology that is not store in the blockchain or public ledger. the only way you transaction can remain unidentified is by using mixer service. although i have not made any transaction using mixer but from discovery mixer is the only way to make a transaction untraceable.