Is Rome talking about moving the Kiss of Peace to the beginning of the Mass or to the beginning of the Liturgy of the Eucharist?

If moving it to the beginning of the LIturgy of the Eucharist (or the end of the Liturgy of the Word), there is ample precedent. The Kiss of Peace occupies this position in the Apostolic Tradition fo Hippolytus (ca. 215 A.D.). This document is believed to be an account of Roman use. We have evidence that by the 4th century it had been moved in the Roman rite to its current place before the Fraction. In Milan it is given at BOTH places. In the Byzantine rite it comes immediately before the Eucharistic Prayer.

In the liturgies of the Episcopal Church in the U.S. and the Anglican Church of Canada the Peace precedes the offertory, although it may instead be given immediately before the communion.

When it preceds the offertory, it constitutes preparation for the Eucharist. When it is given before the fraction, it constitutes proximate preparation for the specific act of receiving communion. (In some medieval monasteries it was exchanged only by those who were planning to receive communion, according to Archdale King.)

Peter J. Elliott, the rather conservative author of "Ceremonies of the Modern Roman Rite," advocates moving to the Peace to the beginning of the Eucharistic liturgy. I suspect his reason is that if it gets out of hand, it is less disruptive at that place than where it stands now.

In the form in which it is currently practiced, I would like to move it down the street to the Baptists, where it properly belongs.

I actually do like the kiss of peace, although I think it needs some serious catechesis. For a horror story, I visited an African American UCC church and the pastor announced "now it's time for hugs!" The hard part was that I had to offer a handshake before people assaulted me. I'm not a hug person, as any of my current or past girlfriends and family members can attest.

When I was at an Anglo-Catholic African-American Episcopal church, they had the peace last forever. It probably was about 15 minutes. You were expected to greet EVERYONE, but I was wearing my organ shoes and so just walked along the outer edge, getting whom I could. It isn't so much that people liked to gab, but rather that it's an African-American church, and they typically take their time in church - services were usually 2 hours, with the congregation sitting silently and motionless during the whole postlude! In that case, I would say they had the right idea about the peace; they took seriously the dominical command to be at charity with each other before the Sacrament. There's likely cultural things that facilitated that, so I don't think "shake hands with EVERYONE" alone will get that same reverence in a Catholic church. In fact, I would go so far as to say that if the people cannot be catechized, the peace ought to be skipped (as is licit) at Mass. As for catechesis, particularly at rough times in the parish the pastor might remind people of the duty to be "one body" and that if they harbor hatred for any of those present, they should refrain from the sacrament. It seems no one ever mentions I Corinthians.

Unfortunately, the silliness that attends the "kiss of peace" invites even more abuse and lack of reverence in the Mass.

The part that worries me the most is that because, unlike in the EF where the kiss is not only limited to the sacred ministers and is done in an highly ritualized way but also occurs after the fraction and singing of the Agnus Dei, in the OF (at least as it's celebrated in my parish) at the time of the "peace" there are no less than three things going on: the people are glad-handing, including the priest celebrant who leaves not just the altar, but the sanctuary to shake hands with the servers who are off to one side, the EMHC's are supposedly making their way up to the sanctuary where they stand behind the altar during the fraction, and the priest is half-preparing for the fraction (moving vessels, missals, etc.). Meanwhile, we start the Agnus and sometimes the EMHC's aren't even up front yet, so the parade of them continues throughout the fraction and even while the priest is reciting the "Behold the Lamb." Oh, did I mention that they stop at two stations on either side of the sanctuary to squirt hand sanitizer on their hands?

I've brought all of this up to the staff before, as have many other members of the congregation, but the post-modern progressivist liturgistas on staff won't budge. They say it's the best we can do.

*sigh*