sync issues come from
Clipped signals from the IF/video amps (tubes resistors agc).
ineffective sync separater action (tubes resistors coupling caps).
A scope is the most effective way to see where the problem is. You look at the signal going into the separator if good then its the separator that is the issue. If its not correct then trace back all the way back to the video detector.
You can also try signal injection of a composite signal at the video detector, sometimes the levels and direction of the a composite signal from a DVD will correctly drive the video. Care should be taken when ever hooking up testing devises or signal injection to make sure there are no voltage potentials to worry about, and esp if there is a hot chassis.
here is a brief demo of a sync sep checkout with a scope.
https://www.youtube.com/watch?v=Flg5LwBadHI
your schematic shows a plug that connects up the out form the sync sep to the horz and vert circuits, I would start with checking to make sure that the pins are clean and making good contact there.