The truth is there are issues of reverence in the New Rite. The problem isn’t so much the New Rite in of itself but the lack of clarity and room for error in the treatment of the Eucharist that the New Rite presents due to its presentation.
With all due respect, and seen from the lens of a lapsed-and-reverted cradle Catholic, this is rubbish.
Reverence is a matter of attitude, not of rubrics. I worship in a Benedictine abbey of the Solesmes Congregation, where the OF liturgy is extremely reverent. They use virtually
all options open to them at one time or another. For the penitential rite, all three options, with Kyrie in Greek where mandated, and the troped Kyrie, chanted in the vernacular, on occasion during the week. They use all 4 main EPs at various times, from the Roman Canon to EP IV.
No matter which option used, it is reverent and carried out with love and care. There is no doubt about the mystery being celebrated (concelebrated facing the people).
As a chorister in a Gregorian schola that goes from parish to parish each month, I’ve seen two celebrants follow the rubrics exactly, but one being casual, one being very reverent. At one parish, the usual (very casual) priest was away and a priest from a missionary religious congregation took his place. Same words, same rubrics, spoken not sung, yet the missionary priest was completely effaced from the celebration; he was just the instrument of Christ who made himself present. It was extremely moving to watch. It had nothing to do with the form of the Mass, or the rubrics. It was all in his
attitude (and no doubt also his training and the culture of his order).
Now you’re comparing your average parish OF Mass to an EF Mass that is being, most often, lovingly preserved by priests like those of the FSSP, or diocesan priests with a special interest in the older form.
Many people here were around when the Tridentine Mass was the only form (I was, but was too young to remember much), and they will tell you that many, many Tridentine Masses, while faithfully observant of the rules, were far, far from reverent: rushed, mumbled, etc. I understand where that comes from; I chant the Liturgy of the Hours daily, and on a day when I’m rushed to get out the door, I’m nowhere near as reverent as on days when I have time. We are after all, human.
It’s true that the EF has more “rules” or rubrics. But that’s just more to break. If the EF became the only form again, what makes people suppose that all of a sudden the irreverent celebrants who break the OF rules would suddenly become obedient little sheep in the older rite? One can argue “but if only the bishops would enforce discipline”. OK, you would maybe have fewer rules broken. But the attitude would be the same, and would shine through just as in my example of the casual secular priest vs. the reverent regular priest.
The irreverence you sense at regular Masses is simply the sign of our times and of our humanity. The attitudes that give rise to irreverence wouldn’t go away just because we change forms of the Mass.