We kind of miss the point, you say the cross was not how he completed the atonement. I disagree. His death completed the atonement.
Helaman 14:14-15 For behold, he surely must die that salvation may come; yea, it behooveth him and becometh expedient that he dieth, to bring to pass the resurrection of the dead, that thereby men may be brought into the presence of the Lord.
Lots of people died on crosses but only one was our redeemer and savior. To many the cross represents his last and ultimate sacrifice.