In the first place — why would 'blockchain' not survive? Blockchains will work totally fine without mixers and coinjoins and stuff. I'm all for privacy, but everyone simply isn't interested in it whether we like it or not. Only a minority cares about self-custody and being self-sovereign; a lot of people just want to use it just as an investment/hedge.
And yes, even without privacy, Bitcoin and blockchains would still be useful, just as these are still very useful today despite privacy being very minimal.
Still useful, probably--if you do as the powers-that-be command.
Imagine living in Stalin's Soviet Union and only having crypto to transact with. They would obviously ban all private transactions. All your finances would be a matter of public record. This is not some theoretical scenario, but a real-life hazard that we will likely experience in the coming decades.