Padre Pio recited the Rosary during Mass (obviously not while he was offering Mass himself). I fail to see how one can go wrong following the example of Padre Pio.
On the other side of the coin, as you allude to, even the excellent Father Stedman’s My Sunday Missal urged people to “pray the Mass” (echoing Pope St Pius X), and did even say that other devotions should be laid aside towards this end, and this was WRT the TLM. The Stedman missal was intended to be an aid to “Dialogue Masses”, something with which I have no problem, and I frequently “dialogue” large portions of the TLM (but not the Canon) sotto voce myself. But other times, as with the Canon, I am moved to recite, as Padre Pio did, portions of the Rosary. So far as I am aware, there is no magisterial teaching, nor disciplinary directive, not to do so, either in the TLM or in the Novus Ordo.
Just on a personal note, my mother’s present hospitalization, on the one hand, frees me up to go to Mass while she is in hospital, but on the other hand, distance and time issues may force me to miss the TLM (which I normally attend when I am able to assist at Mass) and go to the Novus Ordo. I’m good with that, we have an excellent, liturgically conservative parish downtown. But I shall exercise the same meditativeness and quiet contemplation whichever form I attend.