> I would expect that enabling the privacy mode of a browser also sets the DNT header

It does not; and since private mode disables the existing cache and cookies, there would be little point anyway (and even with DNT off cookies stored by tracking services will disappear as the private mode is ended). Though if you usually have DNT on, the header will still get sent in private mode.

More generally, I suspect having DNT on is a good way to radically increase the uniqueness of your browser's fingerprint, as it is likely a minority of users have it on; personally I ignore it entirely and just use ublock + privacy badger and similar solutions (that includes noscript - efficient fingerprinting is much much harder without JS).