Maybe this is down to my history with comics, but Barrett's manner of speaking never bothered me in FF7, even as an adult. I remember back in the '90s, a lot of comics (X-Men in particular) would depict accent through stylized text.
They'd do it for Moira and Banshee's Scottish accents, Rogue's southern accent, Gambit's weird cajun one-liners etc. And I appreciated they did it, because it allowed me to "hear" those voices in my head. It was the same with Barrett. I never felt he was a lesser human being, or some dumb, uneducated off-the-street person.
Barrett always seemed like someone with a tough exterior, but a soft interior. A muscle-man whose heart and mind were both in the right place. Even having watched replays of FF7 recently, on the whole I feel Barrett comes off as one of the most mature characters in that game.