I’m not sure it’s the exact opposite—at least as I read it. He seems to be saying that the Sacrifice is offered by the priest, alone, but the faithful have a role—but in their place—which is not at the same level as the priest. The people do this through uniting their “hearts in praise”, not by any physical act.
Vatican II places the “presider” on the same physical level as the faithful and is very likely moving toward having the extraordinary ministers physically offer the Sacrifice along with the priest. I’d be surprised if we don’t see that being permitted in the very near future. This is where they are going with it, and it has never been suggested in tradition.
Admittedly, the language is a bit murky on that issue, but he states that “this is a privilege “only for the minister.” We don’t know how involved Bugnini and Bea were when he wrote that encyclical, but I wouldn’t be surprised to learn that they were both very close by.
He’s clearly not saying that the sacrifice is offered by the priest alone (and he goes on to say, in essence, that even if the priest _is_ alone, he’s not really alone, since the Mass is still a public act), but you’re right in that he definitely says that the manner in which the faithful offer the sacrifice is distinctly different from that of the ordained cleric.