The answer to my question came from the late Lutheran theologian Herman Sasse, who wrote in Church and Lord's Supper:
Baptism stands at the borderline of church and world.As the Sacrament proper to the church, the Supper is, therefore, as a matter of principle, not a public event. Thus, the most ancient church celebrated it behind closed doors (Rev. 3:20). For centuries thereafter, the arcane discipline kept the liturgy and doctrine of the Supper strictly secret from Jews and pagans, which is why those writings of the New Testament intended for the general public, like the Gospel of John, make no mention of the words of institution for the Supper.
This explaination made so much sense, I was sort of surprised I'd never thought of it, or heard of it, before. In fact, the day before I read this explanation, I'd been about how important is was to read the books of the New Testament with an eye towards the intended audience. For example, to understand what Paul means in Romans, learn about the heresies plaguing the Roman Christians, because that's who he's writing to. And John's Gospel targets Gentile pagans . More specifically, John squarely targets the Greco-Roman philosophers and theologians, and he lays out for them why Catholicism, and not Gnosticism, is correct. Although he spends some time mentioning how Jesus fulfills the Old Testament prophesies, he doesn't invest most of his energy here, since neither the Gnostics nor the pagans held the Old Testament as inspired (on the other hand, St. Matthew, who writes for a largely Jewish audience, goes to greater length to spell this out).
But the Eucharist was a target of mockery from the Romans. The Roman mystery cult dedicated to Mithras created a rite imitating the Last Supper, and there were frequent accusations that the Christians were cannibals, for feasting upon the Body and Blood of our Lord. So the early Christians were clear that they believed in the Eucharist, but were careful about willy-nilly spilling the beans about the precise Eucharistic rituals. After all, the Eucharist is sacred, and exposing the ritual to outsiders would enable them to more effectively blaspheme It. Even catechumens in the early Church - those who were seeking to join the Church - were excused after the Liturgy of the Word, and not allowed to so much as observe the Eucharistic Liturgy until after they were baptized.
Bonus: Speaking of Herman Sasse, let me close with some more of his insights on the Eucharist from Church and Lord's Supper:
All attempts to build Christian congregations without placing at their center the congregation-forming Sacrament of the Altar are just as much condemned to failure as are efforts to renew the Divine Service without renewing the Lord's Supper. The nineteenth and twentieth centuries' sad experiences in this area only confirm the
lessons of the past. The enormous effort made in the area of church planting during recent generations must be regarded as a failure. It has produced a wealth of societies and files, but not a single congregation.
For Calvin, the body of Christ as a truly human body exists in finite form and must, therefore, after the exaltation, be as far removed from us as heaven is from earth. The Lord's body thus cannot simultaneously be present in heaven and on earth, and in multiple locations on earth. Calvin is not in a position to substantiate these assertions from the Bible, for he did not derive them from the Bible. These are metaphysical statements and ideological presuppositions that he uses to explain the Supper texts.
No sign testifies with such infallible certainty the death throes of a congregation, or a
whole church, as the decline and decay of the celebration of the Eucharist. This is, however, the deadly serious situation in which a very large segment of these Protestant churches of the world finds itself.