Yup, there is a downside to overspinning. Soft jacketed bullets tend to have issues with it, cast bullets skid and show poor accuracy, any imbalance to the bullet will be made worse for accuracy too.. The slightly higher drag from the faster twist won't cut more than a few fps though.
I'm running 8" twist barrels on my 7rum and 7stw; I just don't shoot thin jacketed bullets or cast through them.
I might just buy 1 of the 8" twist 375's just to see whether I need to eat crow on whether the cast bullets will shoot or not.