On a gas grill, I will turn the temp up high like you did, but will turn it down just below medium before I put the steaks on and I will stay there to cook them. If you go inside to do other stuff, you will forget about the steaks and burn them. I seldom use my gas grill nowadays, but use my Traegar instead unless its really cold out, then its the gas grill.
Haven't used charcoal in years, but grew up with it.
Well done here, but will do medium well if I have to. You can keep your medium done steaks