Also, the congregation did not kneel during the Canon until the choir had finished with the Sanctus.  And this created a bit of incongruity, since the priest made it to the Consecration before the choir finished.  At that time, the altar boys knelt down, but the congregation was still standing.  Now, the priest waited to do the actual Consecration until the choir finished, but it was strange to have the congregation standing while the altar boys knelt.

Also, the congregation did not kneel during the Canon until the choir had finished with the Sanctus.  And this created a bit of incongruity, since the priest made it to the Consecration before the choir finished.  At that time, the altar boys knelt down, but the congregation was still standing.  Now, the priest waited to do the actual Consecration until the choir finished, but it was strange to have the congregation standing while the altar boys knelt.

The question is, how serious a thing is it?

I was told that the "synagoga" part ("the ѕуηαgσgυє", but also St. Peter, Judas, Pontius Pilate, etc.) could be sung by a layman or schola in cassock.

Ladislaus is right that liturgically speaking the Gospel is the special privilege reserved to deacons and higher. Especially if the priest isn't also reciting it (liturgically). There is a big difference between liturgical chant and "hymns sung during Mass". Women can sing the latter; never the former.

What would be a step lower would be for a mixed choir to sing the part. I've only seen it sung by men though -- it's like they do know it's liturgical. For example, the singer(s) have to be in cassock -- again, that is why women are excluded.

I know many things are done in an ideal manner at a seminary, but most chapels don't have spare priests or deacons around. Unless the priest is a chant/music nerd (I use that term in a neutral to positive sense), it seems like a lot of priests want some help singing it -- at least the synagoga part.

I thought the SSPX-printed booklet addressed this issue.

Long story short, the thinking is probably similar to the situation with altar servers: Ideally, only a seminarian who has received the first 4 Minor Orders (Porter, Lector, Exorcist, Acolyte) has a right to serve Mass. But since most chapels don't have seminarians available, boys in cassock take the job.

Ideally only a Lector or higher would raise his voice to be Cantor (intoning the first line of psalms, hymns during benediction, or singing lessons, the Martyrology, etc.). But even Fr. Timothy Pfeiffer wanted me to be Cantor for Saturday Compline about 14 years ago at our local SSPX chapel. We had just finished a marriage prep class of some sort with several other couples. You need 2 people to do public divine office: the priest and a cantor. We only had one priest and a bunch of laymen.

But the exceptions didn't end there. I remember being Cantor several times at the seminary, even though I never made it to Lector. For example, at the end of the year a bunch of upper years go on retreat, and liturgically the seminary becomes very short-handed. Ditto for over the summer vacation, when there were mostly "first years" manning the place.

With the exception of Lasance, whose directions are not as precise as those of his contemporaries and could therefore be interpreted either way, Fortescue, O’Connell, Reid, Sheen, and McManus state categorically that people should remain standing until after the Sanctus and Agnus Dei are said or sung, and rightly so, because these prayers are the Ordinary parts of the Mass that the Church has appointed  specifically for the faithful’s active participation.

The Rites of Holy Week by Fr Frederick McManus, published in 1956 by the St Anthony Guild Press, Paterson, NJ, states that, the "Passion is divided into 3 parts: the narration (C for Chronista), the words of Christ (+), and the words of the crowd or of anyone else (S for Synogoga)...The choir (even of lay persons) may take the part of the crowd; the 3rd deacon then sings only the words of individuals (i.e. Pilate, Peter, etc)."