Most of us learned to approach the Lord's Table with a private checklist. Search your heart, confess your sins, make sure you're right with God before the bread comes around, or you'll “eat and drink judgment on yourself.” It's sincere, it's reverent — and it's very nearly the opposite of what Paul actually wrote.
The mistake is understandable, because centuries of tradition have built a great deal on top of a rite that says remarkably little about itself. Start with what the New Testament plainly states the Supper is for.
Paul gives it in one sentence: “As often as you eat this bread and drink the cup, you proclaim the Lord's death until he comes.” Two purposes, and only two — remembrance and proclamation, pointing back to the cross and forward to His return. That's the stated meaning.
Everything else — that grace is conferred through the act of eating, that Christ becomes physically present in the elements, that the bread and wine change substance — is inference layered on afterward. The text does not say it. And when the Reformers pushed back against medieval sacramentalism, most of them still kept the underlying assumption that something mystical happens at communion that happens nowhere else. They trimmed the doctrine. They didn't question the premise.
Paul himself rules that premise out, and he does it with an Old Testament framework he builds deliberately across three chapters before he ever gets to the supper. In the sacrificial system, the priests ate a portion of certain offerings — but notice which ones. They ate the peace offerings, the fellowship offerings. They did not eat the sin offerings, the atoning sacrifices. Atonement was accomplished by a different sacrifice entirely, one the priests never consumed.