If you could choose between two identical experiences, except one couldn't see your data, which would you choose?

Everyone (assuming they are fully informed and have time to consider the decision) would prefer the more private option, all else equal.

That's not to say this edge is very large; if the more private option is even a little more inconvenient, it might not be a viable choice for users.

If some options add intrusive logging, but they are easy to replicate and small, then there will be a competition for ones that have less logging.

It will be easier to create a drop-in replacement for the privacy-invasive alternative, and once the same functionality exists without tracking, users will over time shift to it.