That and there's either an op-amp bandwidth problem or it's the diode. I don't think it's the diode.
Based on my experiences earlier this week with a simple peak detector circuit (just a diode, cap, & resistor), I think it's the diode. We saw exactly the same behavior as in your video, and checked the diode and found it to be the culprit. To check the diode, just hook it up in series with a resistor (i.e., diode followed by resistor) and put your AC voltage across the pair, measuring the voltage across the resistor while turning up the frequency. You'll see the voltage go from a half-wave rectified version of the input voltage to a DC voltage that drops to zero as you increase the frequency.