The hassle of complaints and the possible compromise of "private" information (like what your browser sends) isn't that significant. The pain from complaints is trivial, largely confined to Hacker News threads of no significance whatsoever. The ostensibly private data is protected contractually and legally.

I'm not aware of any approach that both yields useful information and eliminates any possible risk of compromise. Surveys and A/B testing and self-hosted solutions all have the same problems and risks, and generally extra costs to boot.

In short, the cost-benefit analysis isn't all that different from that of a company considering an analytics tool.