That's a false dichotomy. We don't need to run things with our private data on third party servers. There is a way out: browsers have very powerful JS engines

That's a false dichotomy. We don't need to run things with our private data on third party servers. There is a way out: browsers have very powerful JS engines and it is possible to do client-side processing and even store private data encrypted somewhat securely without the service provider having access. But since this concept doesn't lead to profits it is completely disregarded. There are many systems using this approach, like password managers and secure email services. It's not perfect but it's been out there for many years.