I have a little different take on things.....

First.... why does a dog who is not house trained have run in the house without someone watching it? A pup that is in house training mode in my house NEVER is left unattended. If you are crate training then the dog is immediately taken outside (pick it up if you have too) to potty. Once it potties outside then it is brought inside to play (supervised). If the pup is inside and I'm busy doing something else then it is tetherd to me with a rope or leash if the pup isn't ready to go back in the crate. That way, if a mistake happens, I'm there to witness it and make corrections immediately.

In making corrections and letting the pup know it did something wrong, you have about a 3 second window of opportunity (at most 5 seconds). If you don't catch the dog in action within that time frame, it is a useless confrontation to correct him. I personally don't think it matters if the dog sees you clean it up or not. The time for the consequence to sink in for unwanted behavior is past. Any frustration you have in cleaning it up with the dog watching or not watching is wasted energy. The dog may pick up on your frustration but still won't know it's something they did to cause it.

If you want the quickest potty training success, then set the dog up for success instead of failure by never letting it loose without supervision until potty training is learned.