There might be another reason.
I've encountered the problem describe here on occasion. It never bothered me enough to look into the cause in depth, but it's definitely not a cookie issue.
Usually it's just that I was forgetting to click that checkbox that keeps you logged in. Sorry I can't remember what it's called but it's the only checkbox you can click when signing in. "keep signed in" or something. Sometimes my browser would randomly unselected it for unknown reasons.
You could be right and it's a cookie issue, that would do it as well. But when I've experienced the problem described here (I think it tended to happen with safari) it was related to the browser not properly selecting the "stay logged in" checkbox.
And as you mentioned, if you use in private browsing (which only helps keep your session private from other people who have access to your computer by the way, not with being tracked by law enforcement for example). Then you will find yourself logged out as soon as you close the private window.
