But that non-interference comes from exposing all the information it operates on to the public. Otherwise, you wouldn't be able to maintain a consensus.
(if you encrypt your data, then nobody else can see it, but you can't do anything with it on the chain and you might as well store it in S3 or wherever)
But still, you could have a centralized service that involves client-side (potentially homomorphic) encryption and stores (and in the case of homomorphic encryption, does operations with) the data on their servers. So that shouldn't give a privacy benefit from blockchain over centralized servers.
You would get a privacy benefit in the sense that the computation provider would not learn anything about the data you gave them. However you are right that centralized computation providers are equally constrained (ie don't learn anything) as blockchains in this regard.
I fail to see how that's better. And if encryption is the solution, there could be encrypted centralized or decentralized apps.