First, it is not clear to me that angels sing. Many Christians assume they do, but I'm not so sure.
Take the Christmas story in which an angel of the Lord breaks the news to the shepherds that a Savior was just born in Bethelehem (the city of David). Immediately after that announcement,
". . . there appeared with the angel a multitude of the heavely host praising God and saying, 'Glory to God in the highest, And on earth peace among men with whom He is pleased'" (Luke 2:13-14).
Notice the text does not say the angels sang; rather, they praised God and said. Several Christmas carols refer to singing angels, such as
"Angels we have heard on high, sweetly singing o'er the plains,
And the mountains in reply echo back their joyous strains."
"Hark, the herald angels sing,
Glory to the newborn king."
Again, however, the Bible does not clearly and unequivocally state that angels do, in fact, sing.
In Revelation chapter 5 we read:
". . . the four living creatures and the twenty-four elders fell down before the Lamb. Each one had a harp and they were holding golden bowls full of incense, which are the prayers of the saints. And they sang a new song: 'You are worthy to take the scroll and to open its seals, because you were slain, and with your blood you purchased men for God from every tribe and language and people and nation. You have made them to be a kingdom and priests to serve our God, and they will reign on the earth'" (vv.8-10, my emphasis).
Obviously, there will be singing in heaven (and almost certainly already is), but the angels evidently do not sing. Later in Revelation chapter 5, we read,
"Then I [John] looked, and I heard the voice of many angels around the throne and the living creatures and the elders; and the number of them was myriads of myriads, and thousands of thousands, saying with a loud voice, 'Worthy is the Lamb that was slain . . .'" (vv.11-12 NASB Updated Edition).
A few Bible versions have the angels singing (e.g., GWT, GNT, NIV, NLT, The Message, and Tyndale), but most do not.
Perhaps as you suggest, the angels may never sing, because they will never be able to
"sing the song of the soul set free" (This is a partial lyric from an anthem I remember from my youth.)
In other words, only those who have been redeemed by the blood of the Lamb are ready, willing, and able to sing, since they, unlike the angels, have been forgiven of all their sins. The holy angels will join (and most certainly have already have joined) the celebration in heaven one day, but only redeemed men and woman, boys and girls, will have a ready song on their lips for all eternity.
In conclusion, I Peter 1:12 we read:
"It was revealed to [the prophets] that they were not serving themselves, but you, in these things which now have been announced to you through those who preached the gospel to you by the Holy Spirit sent from heaven--things into which angels long to look."
Evidently, the holy angels are a bit curious about the gospel message and all that was entailed in making the good news available to sinners who repent and believe. Yes, they have gladly praised God for millennia and will continue to do so for all eternity, but they are evidently puzzled about what God's salvation is all about! Only the saints of God have experienced the mercy, love, and grace of God.