Yes. And like so many other behaviours in the web-stack, I feel like I'm in a constant fight with my client software to please choose privacy over convenience. So it's worth being aware of where these tradeoffs exist. Especially when I'm writing that client software.

If you're that paranoid about your privacy, then I recommend you choose a user agent whose philosophy on privacy more closely represents your own.

Tor Browser, for example, is highly likely to "choose privacy over convenience" whenever possible with it's default settings.