It turns out that I haphazardly found the solution, which I will now post for posterity…
The view that gets used as the sheet (NSWindow or NSPanel) needs to have a title bar. As soon as I toggled this in Interface Builder’s inspector (Window → Appearance → Title Bar checkbox), I recompiled and the NSTextFields highlighted, tabbed, and accepted input like they would in any other view. Not sure why the title bar makes a difference, but there you have it.