It would be more appropriate to transfer it to the 11th. Then Christmas is not truncated and the day is still observed. For the more stringent Anglo-Catholics, that date still falls within the Octave of the fixed calendar date.
That’s basically what the Armenians do much of the time (although not for Epiphany, for on January 6th they celebrate the Nativity and the Baptism of Christ, which was the universal custom prior to the fourth century, and I’m glad one church preserved it). For most holy days however these get pushed back to the next Sunday.
In many Byzantine Rite churches (EO, EC, Protestant) and some Anglican churches the custom is to do both - a lightly attended service on the day of, followed by Epiphany Sunday or Transfiguration Sunday et cetera. There is also the Byzantine custom of the Apodosis, or after-feast, also commonly called the Leavetaking, for example, the Leave-taking of the Nativity, or of Pascha. This corresponds to the last octave day in the Western Rite, except it’s not necessarily separated by an octave but rather the distance between the Forefeast, the Feast and the Afterfeast is proportionate to its relative importance.
Also in the Menaion, which corresponds to the Propers in the BCP or the Missal, for fixed-date feasts, one will find occasional variable date entries for the Sunday following a given feast. Of course everything is more complex and there are a gazillion rubrics in the Typikon and thus to make things simple, many jurisdictions have liturgists who simply prepare pre-formatted instructions, and these are also available in book form, since manually figuring everything out from the source material is non-obvious. This is something I love about Anglicanism: the BCP is extremely simple, but can be enriched, for example, if one cares about octave days, there are splendid resources like the Directorum Anglicanorum and Ritual Notes which provide that. This flexibility allows for a liturgical unity in diversity.
The one thing I really like about the 1979 BCP was the integration of more Byzantine elements (the 1928 BCP already has several Syro-Byzantine aspects, some inherited from the original BCP like the Prayer of St. Chrysostom and others from the Scottish non-jurors - the epiclesis from the Divine Liturgy of St. James; the additions that I quite like concern some aspects of the Paschal Vigil service and also the inclusion of Phos Hilarion at Evensong, although sadly there aren’t very many Anglican settings of that ancient hymn). Like you I prefer the 1928 in general, but I did like that; some Episcopalian churches have used these to great effect, although my favorite parishes, St. Thomas Fifth Ave and St. John’s in Detroit lean in a more Western-purist direction (actually the latter uses the 1928 BCP; I was shocked to learn they were not Continuing Anglican; St. Thomas Fifth Ave uses Rite I for the Eucharist and for Evensong is basically using English rubrics, since they include both parts of the Preces, and omit Phos Hilarion. However, back when the National Cathedral had a boys choir, and even now with its youth choir, and also the struggling boys choir (it was devastated by the pandemic) in San Diego, regularly celebrated Evensong using Rite I with Phos Hilarion, the differences being to the point that when the former was featured on BBC Radio Three Choral Evensong in 1999, the presenter had to advise listeners that the structure was slightly different from that they were familiar with, whereas this was never required with St. Thomas Fifth Ave (which has a very English feel - indeed T. Tertius Noble was once the organist there before moving to York Minster, and more recently a music director went from St. Thomas Fifth Ave to Westminster Abbey).
