OK, so I did some more testing. If I short the BNC:
with the coupling set to GND, there is absolutely no difference. But if I change the coupling to DC or AC (even with the BNC shorted) the noise floor goes up. Also the voltage on the trace shows zero change when shorting the BNC. So it
does look like the coupling to ground is working.
Weirdly though, now I ran the FFT with a faster time base and it looks different. The spurs are now at -80dBV? I don't know much about this HF stuff so I can't make heads or tails of this.
In any case, that 62.5MHz signal is real, and it's coming from inside the scope.