Actually, the formula for the penitential rite is not general absolution, nor does it replace confession. The Penitential Rite is a means of forgiveness for venial sins (so is an act of contrition, for that matter), but it does not absolve serious sins. That requires sacramental confession to a priest.
General absolution is something else, and, unfortunately, it has been abused in this country. The bishops have had to say that General Absolution only valid when there are serious circumstances (imminent danger of death, plane crash, soldiers going into war, etc…), not merely if there are too many confessees and few confessors. In short, you will practically NEVER find general absolution in the United States. If you do, be very careful. It also assumes that if you survive, you get to a regular confession ASAP.
P.S. There is no sign of the cross after the penitential rite (“May almighty God have mercy on us, forgive our sins, and bring us to everlasting life”). That sign of the cross was from the Tridentine rite, and it didn’t complete the penitential rite, but rather, it STARTED the Indulgentiam prayer which immediately followed the PR. Many Catholics back then would cross themselves at the end of the PR in anticipation of the Indulgentiam. however, the Vatican purposely suppressed this prayer precisely because of the growing concern that many Catholics were starting to believe that the prayer absolved from serious sin. Despite the Vatican II GIRM, many Catholics, to this day, cross themselves at the end of the PR - not even realizing that the prayer that SotC corresponded to no longer even exists in the Mass.