I think I may have found the Jewish context that makes sense of that, the missing piece of the puzzle. It starts with the Jewish understanding of a certain Psalm:
Yet the LORD will command his lovingkindness in the daytime, and in the night his song shall be with me, and my prayer unto the God of my life. (Psalm 42:8)
The Jewish understanding of this verse was that the "night-time" reference was about the song of the angels: "in the night his song shall be with me" was the angelic hymn of praise. And why was the angelic hymn of praise heard specifically during the night? According to the Talmud, it is because during the day, the angels kept silent for the sake of Israel, that God might hear Israel's prayers.
"there are companies of Ministering Angels, who utter divine song by night, and are silent by day for the sake of Israel's glory, for it is said: By day the Lord doth command His lovingkindness, and in the night His song is with me. (Talmud, Mas. Chagigah 12b)
Some helpful footnotes to the Soncino Talmud explain that the angels are silent by day "because Israel utters God's praise by day" and that, "By silencing the angels by day, God shows his lovingkindness to the children of Israel, who are thus permitted to win divine grace by their prayer."
Apparently, when Paul cited that the women should be silent "for the sake of the angels", it may have meant: If even the angels in heaven are silent during human worship that God might attend to it, how much more should the mortals be silent from things that would disrupt it. So if some congregation had an issue with some wives who didn't understand some aspect, they shouldn't be whispering to their husbands during the service for an explanation, but saving their questions for home, and keeping silent "for the sake of the angels" who also kept silent so as not to disrupt the prayer and praise.