So back to the topic at hand,
Actually when Mario is talking naturally and not shouting every sentence, it actually doesn't sound too bad.
I'd love to see Mario talk a bit more. I mean I loved the gibberish talk in Mario & Luigi, for example! No I could go further and say I love the short lines in the Mario Advance series, like "Lucky!" or "Thank you very much!".
So yes, I can see Mario speak a bit more, no need to narrate any story, but I can see charming dialogue.
Mario's voice is not annoying, the main issue is that he only shouts, and that's why he can sound a bit annoying. Mario's voice is nice for speach, actually! Fun and friendy!
Actually when Mario is talking naturally and not shouting every sentence, it actually doesn't sound too bad.