One thing playing Medal of Honor did was prompt my son to ask a lot of questions about D-Day. He was blown away when I explained that men really ran up those beaches right into the machine gun fire as depicted in the game.
I believe the combat engineers were among the first ones in - to try and clear some of the obstacles. More guts than I'll ever have.
Yes the show was on “the History Channel” and if my memory is right they said the Combat Engineers suffered 80% casualties, doing just what you suggested.