I don't intend here to go into the original significance of the Pax at its traditional place in the Roman Rite; in posts some time ago, accessible through the search engine, I showed conclusively that the idea was that a liturgical Kiss concluded and sealed what had just been done (here, in the case of the Eucharist, consecration and oblation). No; I want to put some question marks against the significance assigned to it in the Pauline rite. And in current de facto praxis within the mainstream Church
"The faithful implore peace and unity for the Church and the whole family of men and express mutual charity with each other, before they share one loaf". MR 1969 IGMR para 56 b.
I doubt whether it would be easy to find much patristic support for the thought that Christians ought to be concerned for the unity of the human family qua unredeemed - that is, for humanity before and without Christ. You don't have to be a Calvinist believer in the massa damnationis to have read your New Testament and to know that 'brethren' are brethren because it is Baptism that makes one a co-sharer in Christ's Sonship so that one can cry "Abba" in fellowship with all those others who have been so admitted into His Body.
But my real dubium is about the logical link apparently asserted here between the Kiss and the eucharistic sharing in the Panis sanctus Vitae aeternae, the One Loaf. I do not understand why/how those who are admitted to the Kiss can be excluded from the Loaf ... or why/how those to be excluded from the Loaf are admitted to the Kiss.
Modern custom is that on the Solemnity of SS Peter and Paul, the Orthodox metropolitan who is visiting Rome to represent the Phanar is the first to receive the Kiss from the Pontiff. And indeed, in ordinary parish churches, there can be no doubt that it very commonly occurs that non-Catholic and even unbaptised visitors at Mass will (whether they like it or not) be given a warm greeting at Pax time (even though the paragraph quoted above technically suggests that the Pax is confined to the fideles).
Old-fashioned worshippers are sometimes opposed to the Peace because this silly little piece of faux-friendly hypocrisy interrupts their private piety. I have some sympathy with this, but I feel there are more profound reasons for deploring what have become the accepted customs of the mainstream Church.
Or at the very least, questions which need answering.