I definatly have valve issues,...at least at low rpm. Watch the air bubbles in the output line,...they're pulsing through the line and appear to stop or even reverse direction very briefly. I'm not sure if this indicates valve(s) not opening, or not fully closing.
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Your logic concerning fixed displacement pumps seems sound. I used an online calculator to determine total displacement: Engine Displacement Calculator which is 2.988 CC. At 1000 rpm, regardless of pressure, the pump should be pushing out 3 liters/minute, the problem is maintaining RPM as the pressure increases.
How many rpm was the video? Is the line closer to the camera the input?