For over twelve centuries, from the inception of the Christian Church until the 13th century, the universal, apostolic, unchallenged practice of all churches, East, West, North, South, Latin, Greek, Syriac, Armenian, Coptic, Ethiopian, what-have-you, was to anoint newly baptized infants and children with holy chrism (in the sacrament of confirmation or chrismation), and immediately extend to them the most holy and saving mysteries of the Body and Blood of the Lord at the eucharistic altar.
It makes sense: a fully initiated member of the Body without any sin on his/her conscience (thus, by definition, all baptized infants and young children) is entitled to the source and summit of the unity of Christians, the sacrament of the very Body and Blood into which they were baptized.
As St. Augustine of Hippo said clearly in his Sermon 174, “Yes, they’re infants, but they are his members. They’re infants, but they receive his sacraments. They are infants, but they share in his table, in order to have life in themselves.”
Around the 1200s, for a variety of reasons, the Latin church began to stop admitting young children to the altar, related, partly, to its gradual and eventually complete banning of the administration of the chalice to the laity. Having transitioned a few centuries before to unleavened wafers instead of leavened bread, a dry wafer is understandably not an ideal texture to give to an infant. It also became less and less a practical issue given the infrequency of lay communion (to such an extent that the Council of Lateran IV in 1215 had to mandate at least an annual communion at Eastertide — presumably, even that was pulling teeth for many at the time).
In a previous era of unselfconscious “tradition” with scant evidence at our disposal and no well-developed historical scholarship transcending geographic or church-political biases, it was perhaps an excusable mistake to have thought that “this must have been the way we’ve always done it.” But we now know, as a matter of the undisputed historical record, that the evidence of more than a whole millennium lies on the side of confirming and communing infants and small children — or, to be more precise, of not excluding anyone from communion on account of age or “state of reason” who is not known to be a notorious sinner and excommunicate. (For a comprehensive scholarly treatment of the state of this question, see e.g. Notre Dame liturgical historian Maxwell Johnson’s The Rites of Christian Initiation: Their Evolution and Interpretation, Revised and Expanded Edition, 2007.)
Is it good and useful to try and promote “Eucharistic revival” with pilgrimages, adoration, conferences, and pious talks on the importance and reality of the Eucharist? Relatively speaking, sure.
But if we want to truly give free reign to the Lord to show us the power and glory of his abiding presence, we need to start doing again what he commanded of old: “Let the little children come to me, and do not hinder them, for the kingdom of heaven belongs to such as these” (Matt. 19:14).
Tom's Digest 
I agree that the current method, especially in the United States (baptism at birth, communion at ~7-8 years old, confirmation around 16) would benefit from some changes. I would want to, at least, see confirmation younger and before First Communion.
I hadn’t thought it through to the ancient tradition, but it does make sense.
Relative to what?
These sorts of periodic “revivals” are basically a waste of time. Plus one only “revives” something which is dead or at least near death. In any event, these sorts of things appeal to the more fideistic, emotivist element among the faithful. I mean, how many “conferences” and “pious talks” can you really have about the Eucharist? And for what end? To develop ever more wispy and emotive “piety”? Piety isn’t faith. And relying heavily on piety is, ultimately, a sort of anti-faith.
That aside…
There’s nothing wrong with administering the sacraments to an infant. If a particular part of the church decides to do that, fine. That’s a matter of benign “tradition” and custom. Let people decide what they think is best within reason.
What is wrong- in the Roman church anyway -is, in no particular order, the Augustinian theological superstructure of ever multiplying (and thus ever more vague) notions of “grace”, the idea of an “original sin” with its attendant “inherited guilt”, the harsh dualism that sort of thinking entails between creature and creator, the idea that sacraments actually confer this (or these?) “grace(s)” upon receipt, and last but not least that God despite being eternal, omnipotent and omniscient is somehow constricted and limited to the performance of sacramental actions (which entails, logically, that it is impossible for an infant “to enter heaven” unless a specific formula of words is uttered while water is poured upon his or her brow.)
Needless to say the whole construct above is not only irrational, inhumane and silly but also pretty remote from anything that is part of the gospel. Sacraments are for OUR benefit just like the Sabbath was for the ancient Jews (which of course Jesus is on record pointing out). God doesn’t NEED them anymore than He NEEDED them to perfunctorily observe the Sabbath. But we the faithful “need” them in the sense that we need material visible signs of what it is we believe in. Symbols are a vital part of what constitutes the unity of the “visible church”. And thus, that is a what a sacrament is- a symbol of common belief, not a magical spell with talismans confering “grace”.
Fix that misunderstanding- preferably through formal, “magisterial” writ and subsequent catechesis -and it is more likely that many other dogmatic/doctrinal problems within the Roman church will fix themselves.
… and you posit all these ideas, Matthew, by what authority? “I think that’s silly” isn’t an argument, as though the Church waited for your arrival to set it straight. God isn’t bound by the sacraments, of course, but they do effect what they signify. Plenty of other confessions will agree with you, but the Church pondered/prayed her way to this formula centuries ago, and if she’s wrong, then no argument in the world makes sense. Indeed, neither does Christianity.