OP you did a great job replacing the capacitors and I would not get discouraged. Many times vintage gear has weird failures with the unique parts. Patience and tenacity is needed.
The transformer, it's only copper wire but it either gets bashed during manufacturing leaving a thin spot, or it corrodes and breaks or both.
I'm assuming the break is on the primary side, with the very thin wires? If you are very gentle you can touch the solder-tinned parts of the leads with your multimeter probes to see if you get continuity there, which would rule out a bad connection at the pins. Any sharp angle or bend is also a place for breaks.
It's entirely possible to disassemble the transformer, unwind it counting # turns and rebuild it. I use my oven or hot air gun to heat up the transformer to get the (glued) core apart.
DiodeGoneWild does it all the time in many of his videos.
Or, find another drive transformer. What's the part number written on it? I don't recognize the monitor's logo but the same part would be used in many different models.
Could look at an IBM 5153 and use that one?