No, the Hail Mary should not be said as part of the General Intercessions at Mass.
No provision is made for the general intercessions to include a prayer said by everyone, such as the Hail Mary or a parish prayer. It would be surprising if the Apostolic See even permitted the Hail Mary as an inculturation experiment since according to the 1994 Instruction on the Roman Liturgy and Inculturation, n 45: “The introduction of devotional practices into liturgical celebrations under the pretext of inculturation cannot be allowed “because by its nature, (the liturgy) is superior to them.” (Footnote: Sacrosanctum Concilium, n 13)”.
In Marialis Cultus, n 48, Pope Paul VI wrote: “it is a mistake to recite the rosary during the celebration of the liturgy”.
The 2001 Directory on Popular Piety, n 2 quotes the 1988 Apostolic Letter Vicesimus Quintus Annus, of John Paul II, n. 18:
“The pious exercises of the Christian people and other forms of devotion can be accepted and recommended provided that they do not become substitutes for the Liturgy or
integrated into the Liturgical celebrations.” (my bold highlighting).
The 2002 General Introduction to the Roman Missal (GIRM) approved for the USA, which can be accessed from
romanrite.com/girm.html has about the Prayer of the Faithful:
“71. It is for the priest celebrant to direct this prayer from the chair. He himself begins it with a brief introduction, by which he invites the faithful to pray, and likewise he concludes it with a prayer. The intentions announced should be sober, be composed freely but prudently, and be succinct, and they should express the prayer of the entire community.”
The priest is to conclude the Prayer of the Faithful, not the whole congregation.