I can't believe you would ride on a tire with a plug patch!
I thought I was the only one who did that.
Anyway, I never trim the plugs flush - they do fine. Mushroom a bit at first, but 50 miles smooths them right out.
And string plugs are defn'tly the way to go. I've been through a bunch of the rubber plugs (the ones with the expensive plug gun) and they do not hold up reliably. If there is a cord pressing against the side of the rubber plug, it will shear off in no time.