[old_tv_nut]

As it doesn't change with tuning, it's some baseband pulse in the video/CRT.
Also, you said the H drive adjustment doesn't fix it.

There are probably dozens of small ways this can occur.

I see this set uses PC boards. It may be worth checking all grounds on the IF, Video, Horizontal (and who knows where else). The horizontal output current and voltage is hundreds of times larger than the video signal, so even a small amount of it can be visible. Also as mentioned, try lead dress. Take any of the flying leads and move them around to see if it makes a difference. Make sure the tuner/IF cable is solidly grounded.

When this kind of thing showed up in a chassis design, the engineers would try random extra grounds and variations in lead dress until they found the fix. Any of those fixes could be disturbed over time by repairwork or grounds going bad.