For years ive been browsing with clear cookies after each session. What you describe with first visit notification is not common at all. And when sites actually do it, its usually very subtle and can be ignored like the notification bar on the top of stackoverflow. I can only name one website where I am annoyed by the lack of cookies and that is when youtube doesn't remember my preferred video resolution. Otherwise browsing with fresh ones has no degraded user experience for me.

Actually this is starting to become worse though after EU introduced the must notify about cookies law last year. The ones who do clear their cookies once in a while are punished by being notified about them time after time while the others will blindly click OK and never see the message again.