Most apps implementation (for example chrome) is to store all profile data in a local store, which then gets synced to the cloud, and to all other devices logged in as you.
If you switch off a privacy option, that simply switches off the saving of that data to the store. In turn, that means it doesn't get synced to the server.
An alternative implementation might be to still save the data to the store (so it's available for local use), but tag it in some way so it doesn't get synced. That proves very difficult though because then the store can become inconsistent, with some other contradictory data in the store in a part which does get synced, put there by the server or another device signed in as you.
And that's not how Chrome works at all. When you turn off sync, it just stops syncing up to Google, it's got nothing to do with local on-disk storage.
I feel like there’s a tipping point where “we have the strategy that data is delivered to/from the cloud” becomes a way to punish users who turn features off for privacy reasons ...