[trinescope]

My own understanding of horizontal sweep circuits is incomplete, but I do know this: The horizontal output tube is used as a switch, not an amplifier. Applying a constant voltage across an inductor causes a linear current ramp to flow until saturation. My own guess is that anything that interferes with that linear current ramp will cause non linearity. It could be a tube whose voltage varies with current at device saturation (tube fully turned on), or some other less than ideal circuit condition. I'll wait to see what others have to say about this.

The linearity or efficiency coil Is part of a tuned circuit, most likely to adjust the switching point so that it occurs when there is minimum loss; at zero voltage or current, like tuning a Class E RF amplifier. Again, I'll see what others have to say here.