Coming from Canada, I'll give you an example of the problem with Google News. PS I'm not french.
1) Clicking the news tab will always default to the US news. Even if Google is forcing the google.ca domain
Badly designed browsers when doing private browsing don't allow for ephemeral cookies.
The problem is that you are geolocated by IP (and yes, it gets this wrong if you are using a VPN into a node in another country - it thinks you are in the other country; not solving this "problem" is intentional on the part of the IETF), and a attempted cookie is set saying "They are in Canada; redirect and use the google.ca domain to serve up the first page". So google.ca shows up.
This geolocation is not repeated, and the cookie is not reset subsequently, since it's a relatively computationally expensive reverse lookup operation; if the cookie is there, it's referenced, and if the cookie is not there, it's not referenced. Then your subsequent request comes in through that first page, the cookie is examined, is not seen, and therefore you get the default, which is the US response.
The proper thing for your browser to do is to set an ephemeral cookie when doing "private browsing"; that is, it allows the "set" of the cookie, but since it's "private browsing", the cookie is set in memory in the DOM, instead of being saved in permanent cookie storage.
So it's happening that way because your browser implemented has screwed the pooch on what it mean when you are private browsing, and just blocks all cookie sets unconditionally. In other words, your browser sucks.
NB: Chrome gets this wrong in "incognito mode", as well, in the other direction; it implements ephemeral cookies into the session, rather than the DOM. Presumably, this is because they want cookies for login sessions to persist across DOMs which involve Google properties. So it's possible for an "incognito mode" session to leak information to outside parties for cross-site purposes. You'll see this with "limited number of views per month" sites, like the NYT and other news sites, where if you use the same "incognito mode" session - which persists, even if you close the window and open a new "incognito mode" window. If you restart Chrome, then the cookies are flushed. It's not clear whether this is intentional or just bad programming.