Israel Couldn't Sing, So They Put Away Their Instruments

Wow! I've never noticed this before.

^{HCSB Psalm 137:1} By the rivers of Babylon—
there we sat down and wept
when we remembered Zion.

² There we hung up our lyres
on the poplar trees,

³ for our captors there asked us for songs,
and our tormentors, for rejoicing:
"Sing us one of the songs of Zion."

⁴ How can we sing the LORD's song
on foreign soil?

I'm familiar with the idea that the synagogue assembly arose during the time of the Exile, with verse 2 above indicating that non-Temple singing became non-instrumental at this time. But when I read this for what it says, rather than for what I'm looking for, I see something I've never seen before.

The captors asked the Israelites to sing songs of Zion, but they were too broken-hearted to do so, so they put away their instruments.

Notice that they did not put away their instruments and then sing non-instrumentally. They put away their instruments because they couldn't sing. Had they been able to sing, they would not have put away their instruments.