"In many ways, qua liturgy, the Divine Office is in principle more important than the Mass"
How can it be so? Isn't Mass the greatest?
But the Mass, traditionally, and though much rare than it ought be, still in religious communities is part of the Office. Formerly Mass was said after Terce, Sext or None depending on the type of day it was, and in the EF pontifical Mass there is a presumption of such a tie in as well.
There is then, liturgically, a precedence of the Divine Office in toto over any of its parts. Mass is the culmination of Christian worship, it should not be atomized from it.
Still, even taken separately, as they have sadly been taken for so long, the Divine Office is precisely the common prayer of the Church that permeates and makes liturgical the whole day. Formerly even at the parish.