Which song are you referring to?

There was no particular turning point. As far back as ancient times we have records suggesting singers were accompanying themselves--Sappho, for example. The church's relationship to instruments was different in different places and changed a lot over the centuries. Some interpretations only allowed for the use of instruments specifically mentioned in the Bible for worship, while others preferred no instruments at all. But since instrumental music was almost never notated in medieval music intended for voices, we cannot know the extent of their use of instruments. Many singers accompanied themselves--or probably did--and since these instrumental arrangements were typically not written down, they must have been either improvised or memorized. Also, instruments could have played vocal pieces, and probably did.

The idea that instruments were "lascivious" (as someone commented above) might be similar to today's condemnation of twerking and raves among the more pious. In the medieval era, like now, there were people dancing the night away to music the church did not consider representative of proper morals. There are many surviving dance songs that would have been sung and danced by groups playing instruments and drinking in an overall party atmosphere. The more pious people would not want those same instruments rousing the spirits of the congregation. There are paintings of angels playing instruments in many medieval churches, so that is a hint that certain instruments were probably acceptable in church (and perhaps together with voices). It is not unusual to interpret Hildegard's sacred compositions for voices with harp or vielle, and she lived in the 1100s.

You must remember that until the late medieval era, all (or almost all) medieval music was notated as a device to jog one's memory--so the minimum necessary was notated. If you wanted to remember a song you know today, you'd probably write down the words first, and looking at it, you'd remember the melody. If you didn't think you'd remember the melody, you might write some squiggly lines showing where the song goes up and down. This is what they did in the medieval times too, and those squiggly lines eventually developed into notation. But for a VERY long time, anything notated was meant to jog one's memory of a piece of music--not to represent the whole thing. So that is all we have to look at from the time period, usually just text and a melody or suggestion of a melody, and our imaginations have to fill in the blanks. It is not considered wrong to use instruments for pieces with no notated part for them; one should try, however, to use medieval music theory and counterpoint traditions to develop instrumental parts (instead of modern rules).

The example you gave of the instruments and choir might be Concertato style which was developed in the 1600s, and featured instruments and voices, usually alternating. This was a groundbreaking musical style but not the first time an organ was used in church.

While I can't pinpoint the song you're looking for, I can answer your question somewhat.

The Medieval period you're referring to is actually a massively long period of time, and you're referring to dozens of countries over centuries of history, with lots of stuff going on in that time. So I'm only going to answer you in terms of the Vatican and Western church music coming from the Vatican.

For the majority of the early chuch history, vocal music was considered more "sacred" and instrumental music more "secular". Singing was considered akin to prayer, and virtually all early church music was forms of Plainchant, of which Gregorian chant is the best known today.

In the chants, texts were almost exclusively prayers, orders of mass, biblical passages, and so on. So, when monks or nuns were in church and saying their Hail Marys, they could do it in song as well as spoken, as sung was considered to still be prayer.

At the same time, you had a huge tradition of musicians who would sing to instrumental music. The Troubadors and Trobairitz were an excellent example of this. They travelled around and performed mostly secular music, often about love or nature.

So, singing to instrumental accompaniment was definitely around, and we have written music from that area, but almost none in a church setting.

As for did it happen at all, there's still a ton of debate on that one. It may have happened, it may not, but it was by no means common, and we don't have a lot of conclusive records one way or another.

However, there are a lot of quotes from church leaders saying how much they opposed musical instruments as being base and evil. St. John Chrysostom refers to "dancing, and cymbals, and flutes, and shameful words and songs, and drunkenness and revellings, and all the devil's heap of garbage", and there are many more similar texts taking about the lasciviousness of musical instruments.

This attitude remained until about the 13th century, when pipe organs started to be introduced in to the church. They'd been around much earlier, but it took some relaxing of church attitudes, and a considerable outlay of cash to get them into churches. From then on, you would see organs, and eventually other instruments become a part of liturgical services.

http://musicumscientiae.wix.com/bkd-klavier-werke#!organ-history/c1q3c http://www.jstor.org/stable/841804 https://www.catholicculture.org/culture/library/view.cfm?recnum=9134 https://archive.org/stream/femaledisciplef00smitgoog/femaledisciplef00smitgoog_djvu.txt