Problem 1
It doesn’t work because you’re using new_weber but you’re registering news_burner, and they aren’t the same
So instead use:
[news_burner]
You only need a closing shortcode if it’s containing content which that plugin does not.
Problem 2
Shortcodes don’t work in text widgets out of the box, you can add that functionality via a plugin though, see this question/answer:
Shortcode in Text Widget not working
This is why it works with do_shortcode but not in widgets. It’ll also work in post content