[old_tv_nut]

Look at the grid-drive version. Without the resistor in parallel with the diode, it is similarly indeterminate. But, that resistor is not supplying current, it is draining it, just as the CRT does in the cathode drive circuit. The cathode drive circuit is not indeterminate if there is some beam current, only when the picture is completely black. Fixable the same way as the grid crive circuit with a resistor across the diode to drain a little current when the CRT does not.

The diode does not block the average current, it supplies it as a pulse of current during sync, putting a charge in the coupling capacitor that then supplies the beam current during active video.

Think about the flyback high voltage supply: same thing. The HV rectifier only conducts during retrace, but charges a doorknob capacitor or the CRT dag, which supplies the beam current during active video. The rectifier does not need to conduct during the active scan while beam current is supplied. In the case of the HV supply, the other end of the capacitor is connected to ground, so its voltage slump between pulses is in reference to a DC level. In the case of the DC restorer, the other side of the coupling cap is not connected to ground, but is connected to the video output. So, it holds a fixed (slightly slumping) DC difference between the video output and the driven CRT element. That DC difference is reset to the desired value every time the diode conducts.