Today is a favorite day for many Catholics to utter an annual complaint that today isn’t an obligatory Mass attendance day. Frankly, except for the whining, I’m in agreement with this line of thinking. As a start. There’s more (or should be more) to just strong-arming the laity into the Church.
Acknowledging that Church attendance will be somewhat less than on Sunday, the following would be a minimum obligation of the parish:
- Epiphany Vigil with an early evening Mass, if a priest is available. Or maybe Vespers I. Followed by a light dessert reception.
- A morning Mass, certainly. A breakfast following, especially is there’s a school attached.
- Speaking of that school, attendance but no classes. A retreat day and festive activities, including a Midday prayer and a nice lunch, provided.
- An evening Mass followed by a nice dinner. Cap the day off with Night Prayer about eight or nine-ish. Unless the parish considered an adults event. In which case, Night Prayer before midnight.
I think if a Church can’t put the festival back into the feast day, then obligation is quite off the table. It’s far from being a matter of the day, the readings, and “depriving” the laity of a narrative they know as well as any part of the Christmas story.
If you’re not doing the minimal festivity, like this, my feeling is your complaint card has been pulled. As for me and my house, it’s the baptismal anniversary of the young miss. So we’re going out to eat tonight.
Top image, James Tissot, French painter.
Catholic Sensibility 
Another possibility is to keep the traditional Epiphany Propers transferred to a Sunday, but to use the “Second Sunday after Christmas” Propers on the date of January 6th (probably with some modifications) and have a “little epiphany” on traditional date.
For example, the Collect from the 2nd Sunday is a lovely text w/ Epiphany themes.
Almighty ever-living God,
splendor of faithful souls,
graciously be pleased fill the world with your glory,
and show yourself to all peoples by the radiance of your light.