The Sprinkling Rite (i.e. the Asperges Me, or Vidi Aquam in Paschaltide) used to be part of the Solemn Mass. The GIRM has this to say about the place of the Asperges in the Pauline Mass:
The Act of Penitence
51. Then the priest invites those present to take part in the Act of Penitence, which, after a brief pause for silence, the entire community carries out through a formula of general confession. The rite concludes with the priest’s absolution, which, however, lacks the efficacy of the Sacrament of Penance.
On Sundays, especially in the Season of Easter, in place of the customary Act of Penitence, from time to time the blessing and sprinkling of water to recall Baptism may take place.
This perfectly vague language is what has contributed to the disuse of the practice, I think. The Pauline rubrics do not distinguish between low, sung, high, solemn high, and pontifical Masses. This, I think, is actually a
good thing in that beautiful rites like the Asperges have the potential to be performed
each and every Sunday during at least one Mass. But wouldn’t ya know, the perfectly vague and dumbed-down English language translation of the liturgical directives has once again made it possible for priests and bishops to throw out elements of the “old rite” that are considered to be “too solemn” for “today’s” mass-goers.