If you use a blanket rule like that, then the most likely outcome is that you'll stick to your guns for a while, but once the benefit gets large enough, you'll give up and stop thinking about it at all. Which, in the long run, results in the worst outcome.
It's much more effective to do a lightweight risk analysis and set some sort of standard for what you're ok with given different levels of benefit. For example, if you're generally ok with a privacy policy, maybe you're fine sending your image to random experimental sites that don't have any obvious motive to profit from abusing your data. (Which is actually not quite the case here -- the site in question is specifically thinking about privacy risks, and so probably ought to be given a better prior than a random site.)
Then again, I'm a total hypocrite. If I were to do my own analysis, I would not be ok with having a live mic potentially sending any audio it picks up to a company I already distrust. Especially when their privacy policy is not at all reassuring. And yet, I can see the Echo from where I'm sitting...