The Shaykh’s clarification that repeating the greeting to every person in the gathering is an innovation.

The Shaykh : Indeed, among the innovations that have occurred in the present age, and whose innovativeness can only be judged by one who is well-versed in the Sunnah of the Messenger of Allah (peace and blessings be upon him) and the righteous predecessors, is this innovation: when a person enters, he greets and shakes hands, and the completion of the greeting includes the handshake. Although this is a weak hadith in terms of its chain of narration, its meaning is correct in terms of its content; so he shakes hands, but the error is that he gives a greeting to every person he shakes hands with. "Peace be upon you," "Peace be upon you," twenty people in the gathering, twenty greetings. This is an innovation. Rather, the Sunnah is that when a person enters the gathering, he says: "Peace be upon you," one single greeting. Then, if it is possible for him to shake hands with those present, that is better and more lasting. And if it is not possible, he has fulfilled the obligation of offering the greeting.
As for the handshake, it is a recommended practice (sunnah). As some of the Companions said: "We never met the Messenger of Allah (peace and blessings be upon him) without shaking hands." But adding to the verbal and practical Sunnah is the innovation itself. Therefore, the one who knows that "adding to good is good"—no, this is not an established saying, because if it were good, they would have preceded us to it. This is advice I offer on the occasion of your coming to us, and your smiling in the face of your brother is charity. Welcome.