One of the most useful measurements you can do, when you get your scope, is to also get a x100 probe. These are rated generally to 1500V to 2kV, not expensive, many are on ebay. Then it becomes safe for your scope's input circuits to examine the voltage waveform on the collector of the horizontal output transistor (HOT), because the appearance of it gives many clues as to what the trouble might be.
One thing about H output transformers (or Flyback as some call it) the operating modes are quite different from a conventional transformer.
In a conventional transformer as the primary voltage cycle progresses, power is transferred fro primary to secondary circuits. In the case of the Flyback transformer is it a multi stage process, where at times, significant energy is returned to the power supply. This is how the whole arrangement gets sometimes called efficiency line scanning, and why the damper diode/s in the collector circuit of the H output transistor get sometimes called "efficiency diode.
The operating sequence goes like this:
Starting with the CRT at the beam center of the CRT's face, there is no stored magnetic energy in the Yoke or the Flyback. At this point the H output transistor (HOT) is being hard driven ON by its base circuit and is acting as a switch. It has switched the Flyback primary and Yoke circuit across the power supply's terminals. Therefore the current in the Yoke and Flyback circuits starts to rise fairly linearly and increases linearly of the short time from of around 32uS until the beam is deflected to the R hand side of the CRT's screen. The rate of rise of current is frightening, at many thousands of amps per second, but because of the short time the HOT is conducting collector current usually only rises to a peak value of 1 to 4 amps, depending on the design.
In any case, when the beam gets to the R hand side, the HOT is actively driven into an off state by the the driver circuit. Now the stored magnetic fields in the Yoke and flyback start to collapse (much like a car's ignition coil when the contact breaker opens). A high frequency high voltage oscillation occurs, this is the high voltage pulse that you see on the collector of the HOT, but only a single 1/2 a cycle of it is allowed to occur. Generally it is around the 50 to 70kHz vicinity, so the flyback period is around 10uS or less. This flyback voltage spike is transformed up to supply low power, high voltage circuits for the CRT's final anode (EHT) and often some if its other electrodes.
Not too much power can be taken during flyback time or it damps the pulse. At the middle of the flyback pulse, the beam now travelling Right to Left is in the center of the screen again, the flyback voltage has peaked, and at that point all of the magnetic energy that had accumulated in the magnetic field of the yoke & flyback, on the R side of the scan, has been transferred to the self capacitances of the flyback and yoke and any parallel tuning capacitances that are there on the flyback's primary.
In any case, at the end of the flyback pulse, some interesting things happen:
The magnetic field in the yoke and flyback is now reversed compared to what it was when the CRT beam was on the R side, so the beam is on the left. The electric field of the charged capacitance's that peaked at the screen center has given all of their energy back to the Yoke and Flyback's magnetic field. Then the oscillaton on the HOT's collector tries to swing the collector voltage below ground. When that happens the damper diode conducts, clamping the collector voltage close to ground. This kills (damps) the oscillation out (which is why you only get to see a 1/2 cycle of it) The stored magnetic energy is then injected (returned) back into the power supply in a fairly linear manner to scan the beam from the left hand side toward the screen center and then the cycle repeats. Generally the HOT has already been switched on before the beam is at the screen center.
During the scan time (between flyback pulses) though, the flyback can act as a conventional transformer, where energy is extracted then, for example to run the CRT's heater or make auxiliary supplies to run things like video output stages. It depends on the polarity of the secondary windings and their diodes as to which part of the cycle energy is extracted from.
The point being, that if you examine the collector voltage of the HOT, if all is well, the collector voltage should be flat, close to zero most of the time, with the flyback pulse in between. When things fail like inadequate HOT drive, shorts in the transformer etc, it can be diagnosed from the appearance of the waveform on the HOT's collector. This is where your scope will came in handy, but in color sets, the voltage there can peak to over 1kV, so get the x100 probe too, or you might kill the input Fets in your scope or damage the input amplifier or attenuator system in the scope.