What I would do in Incognito, etc. would depend on the website/app and use-case. The point was that handling that on a case by case basis makes more sense than the suggested silent (and I would say unexpected) behavioural change.
I'd say that if the user is using a private browsing mode, they are explicitly indicating to the browser not to store anything, and they will know that will happen. It's the whole purpose of the feature, so I don't think it counts as "silently" doing anything. I don't think there's any reason to then potentially break web pages that (quite reasonably) expect storage APIs to work.
Right. It's like "silently" removing session and non-session cookies when the windows closes. That's the whole point.