re: Wouldn’t the game have been tied anyway?

No. Had it been a base hit and let’s say the runner on second had been thrown out at the plate for the third out, then the game would have been tied because the run scoring from third would have counted. But with a straight ground out, it would not matter if that runner on third crossed the plate 30 minutes before the out at first. Game over at 7-6 with the ground out.