Some more debugging info, reinforcing my hunch that this may be a buffering problem: Today, for the first time, suddenly I saw Classic Spectrum Analyzer working in the Wacup main window when playing a 16/44 flac file. The track ended, and when the next track started playing, Spectrum stopped working.
HOWEVER, this time I moved forward/backward within the track, either with the cursor keys or the mouse, and Spectrum started working again! Naturally, when the next track starts playing, Spectrum does not work.
So I started testing some of the WASAPI configuration options. Starting from your Defaults for all tabs, it appears that what fixed the problem is setting the Device Period to 'Minimum' instead of the 'Default', in Buffer Options (third tab). Skipping tracks, torturing the playlist, did not affect the effectiveness of the fix. I am happy!