For example a live chat widget with a link "powered by + LINK TO THE SITE" . I've tried to view source and CTRL + F to look for the backlink but can't find it, but the link is clickable and will redirect you the site. Is it being counted by google as backlink? Does robots read it as links even if its not found on source?

If you can't see it in the source code, then it is most probably injected in the DOM with javascript.
Thus it is not counted (because not even seen) by google.