Sure, you can ignore your stutter and go on but it won't give you any real confidence because the fact is still there, I've done this for 20 years. This mindset is just a basic defense mechanism our brain uses for coping. If anything, I think "feeling it" and having a conscious stutter gives more confidence because then you don't feel so out of control. Try forcing yourself to stutter on some words and you will know what I mean. I think that what you see in the video is focus and discipline. 20min conversation like that is about what my limit is before I get too exhausted and start slipping. You can also notice that Ariel does it more further it gets. However with practice and proper techniques (like from speech therapy) this becomes easier. It's hard work but it's sure as hell worth it, especially if you're still young.