The voltage readings on the HOT tend to point the finger at excessive drive.
This would jibe with the waveforms you show the more I think about it. The clipping at the top suggests diode action of the control grid caused by its being driven well into conduction. There is some of this happening to a degree in normal operation, as that is how the HOT develops the negative bias that keeps the cathode current in check. In cases of horizontal oscillator failure the HOT would melt down completely. But it would seem that the reverse is happening here, as your cathode voltage reading and the control grid negative reading reflects little current flow, as if the tube is conducting only during a very small part of the waveform cycle. If that be the case then one would think that the width of the raster would be affected, and that one side of the raster would actually show a brighter edge, reflecting the beam lingering there during the HOT's cutoff. Just the same such condition would cause lower output from the HOT, as it is only conducting for short bursts. Attempts to further increase drive will only make matters worse.
It seems to me that you need to be going in the other direction, if possible.
Do you actually have a video program source or a test pattern in your testing? Or are you just working with a white raster or snow?