As mentioned by several others, the sign of peace has been a part of the liturgy for a thousand years, and in its current location within the Mass. When the priest says, “May the peace of Christ be always with you,” that is part of a long liturgical tradition.
If we then turn to our neighbors, friends, family, or other parishioners, and confirm this by saying “the peace of Christ be with you,” there is nothing in that liturgical moment that by its nature particularly calls for kisses, hugs, or even handshakes. It is simply an extension of the rite of peace just completed by the priest.
It seems to me that the rite can be reformed to its original meaning without being moved. But since its liturgical meaning seems to have been lost in the past 40 years or so, that may be hard, though not impossible, to do.