This stuff is supposed to be fun, it's an adventure repairing vintage gear and keeps your skills sharp
There are many ways to fix equipment - keep replacing parts or troubleshoot each stage to the component level, start the end, or the middle or the beginning stage.
I was looking at the three sections responsible for the H-sweep: Osc. V4/V5, H-amp V6, H-CRT driver V7 and we need to have some idea what stage is not working. Your DCV measurements seem to show everything is reasonable.
So one approach is start in the middle, see if the sweep amplifier has good output voltage, comparable to Paul's scope.
OK, no probe but no problem to poke around in the horizontal section as long as what you are using is touch-safe. Or just power it down and connect the test lead, then power on and measure, every time.
Can you fix the V/div switch wiring with R1, R2, R3 to avoid confusion. Everyone note this scope vertical section is only AC-coupled, so DC offsets do not get in the way.
I would do a measurement at the top leg of the Horiz-Gain pot R52, we need to know how many ACV is coming out of the sweep amp buffer V6A .
When you have the HOR/FREQ selector in LINE SW position, you are not even using the sweep oscillator. It's just a sine-wave 3Vpp FIL-B coming into the horizontal amp, so completely bypassing the sweep oscillator V4, V5. This leaves only V6 and V7 stages to measure.
Using the Vertical section, can you look at:
V6A output pin 3 or at the pot at R52. Should be around 3Vpp?
V6B pin 8 plate H-amp output. Should be around 30-60Vpp?
Biggies are V7 plates pin 1, pin 8 should have lots of voltage swing there but we don't seem to have it.
With blanking disabled, C124 leg lifted I would have expected to see a retrace line, the time when the beam zips back from right to left in readiness to do the next sweep.
If you're not wanting to make measurements, then carefully retest the all parts in the section - R260, R261, C230, R151, R254 etc. without assuming any of them are good.