I have been doing what you suggested and yes it does give me a single sweep when I engage the RESET button and then press it again to release, or as you suggested, switch the selector to LINE and it does a single sweep off the screen and then back to where the original dot is.
In regard to the transistor swapping, one of the first things I did here was swap them all one by one and try it to no avail. I ran through that again on the ones you suggested and no change on any of them.
One of the things that may help is that I am having the exact same problem with both channels, but I do not know what is common to both? Seems like a clue but I am in way over my head on this thing and don't know where to look from here.