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Well, I agree the first part looks dubious. I got it off a surname site. It would be good to know the language (not just 'Celtic') and an authentic spelling. The second part is from the Oxford English Dictionary, so that is not for me to doubt. It is not contradicted by your post, where it says that Irish bragáireachd (the only word in your post with a similar meaning to modern 'brag'; I ignore the others for now) comes from the English language! So this Irish word is even less old than the English, medieval, 'brag'. In short, if it really is an ancient name, any simillarity between the meaning of modern 'brag' and the name 'Bragg' must be coincidental.