When I check chat in the output section, the 'subsection' display is greyed out and I am unable to pick anything regardless of what option I'm choosing.

Is this on retail or classic, and regular releases or alphas? (We're in a moment of some major differences between release/alpha.)

Wrath Classic, release version!

Having looked into this: that's how it's supposed to be. This is a config UI that LibSink-2.0 provides. "Chat" just means to print something to the chat frame, not to actually send it as a message to others. The dropdown is there in case you have an addon installed that registers itself as an output for LibSink and needs to give it options.

(It'd also be meaningful if the "Channel" option was available, which does send an actual message to others -- but Blizzard restricted that so it can only be used directly in response to a hardware event, so I can't have that enabled in SilverDragon.)