> Obviously Google still gets some information, but we do all we can to limit this.
When you cause a request to be made for ReCaptcha, it seems that you're leaking enough information to (in many cases) link a possibly-pseudonymous Protonmail account to an identifiable individual.
(For example, even if you leak nothing else than times that individuals identifiable by Google logged into unidentified ProtonMail accounts, Google can already see various external activity of specific ProtonMail accounts, and you've given them temporal correlations between activity of pseudonymous accounts and logins by identifiable individuals. That's not the only example, but even that alone seems a significant risk.)
And it's seems to be a real risk: Google is in the business of doing things like that, has a track record of doing things like that, and presumably is more than capable enough of doing it some more.
> but at most a percent or two of users are seeing CAPTCHA at any time.
That sounds like a lot. And the "at any time" sounds like an even higher percentage of users are potentially being compromised by the use of ReCaptcha.
> we don't like it either
I'm not yet convinced that this is the least of all evils. And I don't know how much you have to dislike it before you decide not to do it.
For persuasive effect, is it helpful to imagine the reaction of your philosophical adversaries, when they heard that ProtonMail was using ReCaptcha? I just imagined some of them laughing derisively or incredulously. I don't say that to be mean, but I don't understand the rationale for using ReCaptcha, and I want to emphasize that it seems to be a problem that threatens ProtonMail's raison d'etre and/or brand image.
(BTW, I'm assuming this ReCaptcha choice isn't due to legally-compelled cooperation in unmasking specific accounts -- in which case I wouldn't say anything -- since, in that case, I expect you'd find a way to comply without misrepresenting the rationale to everyone else. I've seen ProtonMail thinking ahead to avoid related conflicting obligations and assurances.)
(BTW, I'm speaking here of Google as an adversary of your customers, and therefore of you, only because that seems to be how your product is positioned, and why you have customers at all, rather than everyone just using GMail. I'm not saying that Google is bad; only that I think it should be considered an adversary from your perspective.)