Specifically again: should laity who come from the diocesan Novus Ordo structure into Traditional Catholic groups take an abjuration of error.
Note: I was raised in the SSPX so I have no clue how different groups in Tradition have handled this in the past.
No they shouldn't.
They are presumed to be of good will, which is why they find themselves at a Trad chapel. Anything else (on their conscience) can be handled in the confessional.
Dimond Brothers and other whack-job types are into "abjurations" because they fail to recognize that the Crisis in the Church is unprecedented and causing untold confusion among the Faithful. But they also think that novus ordo Catholics are like protestants
simpliciter, or to use modern language, that they are LITERALLY protestant, and so they need to do an adult Baptism with a profession of Faith, etc.
A 40 year old Novus Ordo Catholic who was brought up in the Novus Ordo and showed up for Mass at an SSPX chapel last Sunday did not BECOME Catholic. You might say he "discovered Tradition" but he was already Catholic.He's behind? Sure. Damaged goods? Sure. Has a lot to learn now? Of course. But you have to be accurate and admit that he was Catholic, and you can't PRESUME some kind of malicious heresy just because he comes from the Novus Ordo. The Novus Ordo is a MIX of the Catholic and the protestant/masonic/other, remember. You can't know which part they lived for, and which part they hated or tolerated. You have to presume the best about human beings.
There are Novus Catholics who have completely lost the Faith; I wouldn't even call them Catholic. Biden or Peℓσѕι, for example. But guess what? You don't see them showing up at a Trad chapel, do you now? They would stop being Catholic if the Church was restored to Tradition. There are some who ARE ONLY THERE BECAUSE of the protestant/worldly/humanist/sentimental elements. These are they who do not have the Catholic Faith simpliciter.
More sane Catholics realize that Novus Ordo Catholics are often completely Catholic, they just need to find a good (traditional) Mass and the guidance of a good Traditional priest. You can't exaggerate the situation or over-simplify. Over-simplification equals error, just remember that.
To be honest, "abjurations" are in the same category as "if you've been to a rival chapel, you need to talk to the priest before going to Communion" -- in other worse, extremist (making dogma things that aren't dogma) Sedevacantist bullshit. Not all sedevacantists or their chapels deserve my criticism here. But many do. And it's those I'm aiming this criticism at.
Fr. Cekada and many priests in that corner of Sede-land, and the Dimond brothers come to mind. Both are obsessed with making "their" followers exclusive to them, getting them to formally abjure things they really don't need to, etc.