In my case, I suspected the problem was that the TV had suddenly stopped accepting volume commands from third-party sources given that it wasn't only the Siri Remote that wasn't working, but the iPhone's remote app (which you access through the Control Center) had all volume functions grayed out as well.
I ultimately fixed the problem by unplugging both the TV and Apple TV for a couple of minutes while disconnecting the HDMI cable at both ends (simply restarting the TV hadn't helped). After reconnecting everything, the Siri Remote now works; however, the volume controls on the iPhone (which I hardly ever use anyway) are still grayed out.