I don't mean this as a personal dig --- not at all --- but are you a Catholic?
I ask because this happens all the time at Catholic funerals. Somebody wants a clearly non-liturgical song at the funeral Mass (say, "Danny Boy" or "Thanks for the Memories" ) and the priest legitimately says, "No, this is a Liturgy of the Church. It is not something designed by me, or you."
And then the priest should helpfully suggest--- as this one did --- that they do the requested song at the funeral home, the parish hall, at the cemetery or at the post-funeral luncheon. These are the appropriate non-Liturgical venues.
This can be explained and arranged graciously. What people do not understand is that the Liturgy is not the "property" of ether the priest of the family. It can't be totally customized according to preference. It is a solemn worship-centered act of the Church.
Or at least it is supposed to be!
That’s a valid issue, if the church expectation is that a fixed format be followed. That is a church discipline thing. Other churches have other disciplines that they keep. Some won’t play a musical instrument in the service, and bringing one in would get you tossed out.
Of course if I were considering such a thing, I’d study the format, to see what really could and couldn’t be fitted. And if it couldn’t, conducting the extra activity (in line with Francis’ “If they are of good will”) after the end of the official liturgy sounds like a winning formula, though. And that’s what they did.
And no I'm not a Catholic, but have been to enough Catholic weddings and funerals that I think I understand the tradition.
Btw, I can't stand "modern" performances in Protestant churches. Guitar strumming, drum circles etc. That silliness has helped to drive away the congregations of the old main line churches, leaving a vacuum that is now filled by the un-Godly Left.