You're right that Satoshi discussed about privacy in the Bitcoin's whitepaper but that privacy wasn't meant to be something like the one that the mixers give. There were no chain analysts back then and Satoshi mentioned something that was acceptable as part of privacy in those days.


It was directly mentioned in the Bitcoin white paper that public can see who is sending an amount of Bitcoin to whom but they won't be able to link that transaction to anyone. I believe these days it's not hard for authorities and other chain analysis originations to determine that who made the transaction and linking it to an individual isn't that hard anymore for them.

The mixers were playing their part to add that privacy element to the transactions and surely they had achieved their goals because linking of those transactions with someone was pretty tough when the transaction went through a mixer but some of the mixers were doing shady things as well which were noticed by the authorities and that become the reason for seizing of mixers.