It really doesn't make sense for Normal to be super effective against anything since it's the "typeless" type. Normal being SE on Fairy makes about as much sense as it being SE on Dragon.
Logic in Pokemon isn't always, ya know, present. Fairy being SE against Dragon doesn't really make sense to me personally without stretching for some sort of explanation, so I could see other type effectivenesses not making complete sense either. Game Freak seems to be focusing on balancing the type chart more than ensuring type interactions make sense, at least in my opinion.
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Normal is already a great type, it doesn't need another strength.
It does not have a strength unless the strength is to be neutral and have a broad move pool but dragons also have a broad move pool and they own. Normal is weak to fighting and cannot hit ghosts and in return ghosts cannot hit them either. That is a negative for resistances.
Having it be super effective against something would be kinda nice.