I realize it frustrates some people to no end…people who frankly are overly scrupulous in liturgical matters…but there is enormous leeway in the rubrics of the Novus Ordo in certain aspects, including in the so-called Bidding Prayers.
The “Hail Mary” can be used here, and so can plenty of other things. It’s a part of the rite that allows tremendous freedom, by design.
Historically, one can see why this rite fell into disuse organically over the centuries.