This talk by LDS Apostle David A. Bednar entitled 'Always Retain a Remission of Your Sins' is a great compilation of LDS scripture showing how the sacrament ties into the essential saving ordinances of being baptized and receiving the Holy Ghost, and the continual need for repentance. It states that there's nothing metaphysical happening with the physical emblems but the change happens in our hearts and wills.
Please consider that the emblems of the Lord’s body and blood, the bread and the water, are both blessed and sanctified[...] The ordinance of the sacrament is a holy and repeated invitation to repent sincerely and to be renewed spiritually. The act of partaking of the sacrament, in and of itself, does not remit sins. But as we prepare conscientiously and participate in this holy ordinance with a broken heart and a contrite spirit, then the promise is that we may always have the Spirit of the Lord to be with us. And by the sanctifying power of the Holy Ghost as our constant companion, we can always retain a remission of our sins.
This expands on the scripture given to Joseph Smith in Doctrine & Covenants 27:2 which you alluded to,
that it mattereth not what ye shall eat or what ye shall drink when ye partake of the sacrament, if it so be that ye do it with an eye single to my glory—remembering unto the Father my body which was laid down for you, and my blood which was shed for the remission of your sins.