Then we know your sweep generator is probably not the issue. X-Y mode bypasses the sweep generator.
I would make sure this assessment is correct and check for the presence of the sweep signal. I would then follow the sweep signal path on this circuitry (with the scope not set to X-Y mode) - from the service manual:

Somewhere along that path, the sweep signal gets lost. If you find where is it that it gets lost, then you'll have found the problem. The problem is, then, how are you going to follow it. Normally, one'd use an oscilloscope, but you said you only have this one. Well, I can think of some options.
First - you could press a sound card into service in lieu of an oscilloscope. Admittedly not the best thing in the world, and there's a chance you could damage your sound card/computer if you do something wrong (i.e. stick it into a voltage higher than 5V without attenuation), but it could be done.
Second - in the olden days, when oscilloscopes were a luxury out of the reach of most repair technicians, they'd use signal tracers/injectors. For this, you'd set your scope to X-Y mode again, inject an audio range signal in the X channel, and then use the tracer to trace the signal from the channel input to the deflection output using your hearing. It's basically an audio oscillator and an audio amplifier. You could easily make one yourself.
Third - follow the sweep signal using a frequency meter, or a DMM that has one. You'll have absolutely no clue whether the signal looks like it should, you'll have no idea of the amplitude, but it's better than nothing at all, at least to check whether the signal is where it's supposed to be.
Keep in mind that the deflection output is in the order of hundreds of volts. Take precautions and attenuate the input of whichever method you choose (if any) accordingly. Read the detailed schematics in the service manual to estimate what amplitude levels to expect in each stage.