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.