It seems two completely separate things are being conflated in this thread:
1. the "Dialogue Mass", a low Mass where the people speak the responses out loud normally said by servers. This is considered normal and unremarkable among European traditional Catholics, but has never caught on in the English speaking world. Trying to impose this practice on English speakers in the U.S. or U.K. where it has never been a tradition would be a bad idea for all sorts of reasons.
2. the practice of people singing the responses and all or part of the Ordinary (Kyrie, Gloria, Sanctus, Agnus) at a Missa Cantata. This is a traditional practice that pre-dates the "Dialogue Mass." It was common in the Middle Ages, started to decline around the end of the Middle Ages with the introduction of polyphony (but never entirely disappeared) and started to be reintroduced in many places in the 18th and 19th centuries, and especially in the 20th century following the 1903 Motu Proprio of St. Pius X, who commanded that "Special efforts are to be made to restore the use of the Gregorian Chant by the people, so that the faithful may again take a more active part in the ecclesiastical offices, as was the case in ancient times."
Unlike the "Dialogue Mass," congregational singing of the Ordinary is widespread not only among European traditional Catholics, but also among English speakers, although it is not universal.
Some people will try to tell you that the congregation did not sing the responses before the 1960s and that it is a modernist/Bunigni innovation. This can be disproved by many documented references from the old days to liturgical singing by the congregation (for example, in Fortescue)
There is a Hollywood movie from 1952 called "When in Rome." An American priest on pilgrimage to Rome has his cassock and saturno (and his money) stolen by a con man, who replaces them with his own clothes. Having nothing else to where, the priest dons the con man's clothes and goes to find the police to report the robbery. They instead suspect that he is the con man and arrest him. He protests that he's a priest. The Italian police officers ask him to prove it by chanting the Preface to the Mass: "only a priest would know how to do that." The priest obliges and the astonished police officers remove their hats as he sings "per omnia saecula saeculorum..." When he gets to "Dominus Vobiscum" and "Sursum Corda" all of the police officers respond to him by singing the responses perfectly. This scene makes absolutely no sense if it were not perfectly normal for Catholics to sing the responses at sung Mass in 1952; so normal that a police officer could sing "Sursum corda" as naturally as he could make the sign of the cross.
As far as having a layman stand in front of the altar rail and try to conduct the congregation as though they were a choir; this does look very silly and completely distracting, and reminiscent of Novus Order "cantors" waving their arms at the congregation. It's unnecessary and a bad idea, even if it is normal in Europe - you can see the practice in of the SSPX pilgrimage to Lourdes in 2014 (at around the 11 minute mark). Better to use the organ as cue to the congregation of when they may sing.
A few comments on this post:
1) That the Europeans consider the dialogue Mass "normal" is because the proto-liturgical reformers were European (particularly among the low countries). They successfully connived, plotter, manipulated, and conspired (as anyone who has read the memoirs of Dom Lambert Beauduin will acknowledge) to prey upon weak and liberal bishops. First obtaining permissions to hold congresses in their dioceses, usually in a monastery to keep things quiet. Then once a plan was formulated, they would seek permission for experimentation. Then the would fabricate reports about how well received the experiments were by the people, and the alleged good fruits obtained thereby. This would cause the weal and liberal bishops to grant further concessions, and their colleagues seeing their brother bishops grant such concessions granted their own, and so the movement -illicit and illegitimate in origin- grew and spread, until finally you had popes saying dialogue Mass.
But it was always based on lies and deception, and contains fully the principles of the Novus Ordo.
English-speaking readers can read the SSPX's own "Liturgical Revolution" by Fr. Diddier Bonneterre for starters.
But let's not pretend the dialogue Mass was "traditional" simply because it was pre-conciliar modernism.
The mutilated 1956 Holy Week of Pius XII (which gained support by the same illicit means of experimentation) was also pre-conciliar, but had only a 13 year history in the Church (with Palm Sunday and the sacred Triduum being almost completely remade into new rites).
2) As for the Missa Cantata, it was to be permitted by exception only, where additional clergy were not present to fulfill the roles of deacon and subdeacon. Why it is celebrated in some places having priests or ordained clerics to perform these offices is inexplicable.
But to say that it is the fulfillment of Pius X's falsely attributed exhortation for the faithful to take part in Gregorian chant is refuted in a recent study by Dr. Carol Byrne, who commenting on this old fable observes:
"If we wish to know the authentic Catholic position that guided the Church throughout History, it was expressed by Pope Pius X:
“The Church is essentially an unequal
society, that is, a society comprising two categories of persons, the Pastors and the flock, those who occupy a rank in the different degrees of the hierarchy and the multitude of the faithful.” (1)
Traditionally clerics or monks composed the church choir - Photo from the New Liturgical Movement
In the context of this two-tier system, it is of the greatest significance that the choir was traditionally considered a class apart from the congregation because its function of singing the liturgical texts belongs to the Bishops and the clergy. In other words, the choir is essentially a clerical entity.
It follows, therefore, that choir members – even though they may be laymen – exercise “a real liturgical office,” for which purpose it was laid down that they should “wear the ecclesiastical habit and surplice.” (2)
As for the other category of persons included in the “multitude of the faithful,” no specific directives were given to them by Pius X, from which we can infer that they were under no obligation to sing the liturgical texts. This is indisputably clear in his explanation that, apart from the singing of the “celebrant at the altar and the ministers,” “all the rest of the liturgical chant
belongs to the choir.” (3) [emphasis added]
The ordinary faithful were, therefore, by definition not included among the singers performing liturgical functions. So, there are no grounds for believing that Pius X had a congregational rendition in mind when he issued his motu proprio
on Sacred Music in 1903.
Even before he became Pope, when he was Bishop of Mantua and Patriarch of Venice, the future Pius X issued documents on Sacred Music. (4) It is interesting that while they are all practically identical in wording and content to the 1903 Latin motu proprio,
none of them mentioned “active” participation of the laity – or even broached the subject of congregational singing. "https://www.traditioninaction.org/HotTopics/f151_Dialogue_68.htm