The "bringer of good news" is بشار with a stress on the "sh" and a long second "a". "Bash-shaar", sort of. (as in Bashar al-Assad, the Syrian President).

"Humanity" is بشر with two short a's.

Hans Wehr (Arabic>English dictionary) has two separate entries for the root b-sh-r: The first one is the root for words like: rejoice, announce good news, joy, glad tidings, bringer of good news (Bashar isn't mentioned in this dictionary, but bashir), missionary etc.
The second one is the root for words like: to peel, to be in direct contact with, to pursue, human being, skin, humanity, practice, direct, etc.

... which suggests that, although the roots appear identical, the two words don't actually have the same root.