Interesting that you equate the Jews' sacrifices with your understanding of the Lord's Supper. You are correct to compare them. Both are incomplete understandings of the "Lamb slain from the foundation of the world."
It is not the physical or material that saves; it is the spiritual.
Then said I, Lo, I come (in the volume of the book it is written of me,) to do thy will, O God. Above when he said, Sacrifice and offering and burnt offerings and offering for sin thou wouldest not, neither hadst pleasure therein; which are offered by the law; Then said he, Lo, I come to do thy will, O God. He taketh away the first, that he may establish the second. By the which will we are sanctified through the offering of the body of Jesus Christ once for all. And every priest standeth daily ministering and offering oftentimes the same sacrifices, which can never take away sins: But this man, after he had offered one sacrifice for sins for ever, sat down on the right hand of God; From henceforth expecting till his enemies be made his footstool. For by one offering he hath perfected for ever them that are sanctified." -- Hebrews 10:6-14"In burnt offerings and sacrifices for sin thou hast had no pleasure.
We are to "remember" His sacrifice; not to repeat it.
Then there was no need for the cross.
Amen. Let all blessing and honor and glory and power be unto the Lamb of God. He alone is worthy to open the scrolls and to read the books of works and the Book of Life. He alone has redeemed us from hell and self, for His good pleasure, according to His good will. Power and praise are His forever and ever.