It's Easy To Fix
Because, as you have a vertical line, you must have EHT for the CRT which comes from the line output transformer. This means the line output stage is working, its just the line scan coils that are disconnected, that will almost certainly be a dry joint on the PCB where the scan yoke leads connect or possibly at the scan coupling capacitor or line linearity coil. It can't really be much else other than an o/c scan coupling capacitor.
How do I know this? well, the college I worked for had lots of Osbornes and I used to repair them when they broke down. Most common fault was the extension card for the double density floppies working loose (these things were carried between rooms regularly which probably explains that), next was the display which could fail to work due to dry joints at the line output transistor connections or by the vertical line due to joints as described earlier.
Once I had to make a new system rom for one (by copying a good one from another machine) as the suspect one was partly corrupt (would boot to the Osborne startup screen but would intermittantly fail to load the o/s from disk) -- that took a while to diagnose.
One other task I had to do was calibrate all the floppy drives so that disks were interchangeable between all machines -- there were quite a few that would not reliably read disks from other machines until this was done.
Most of our Osbornes worked with external monitors (to ease eyestrain on the students).
When carrying machines between rooms I always carried two at a time, that way both arms stretched by the same amount :-) happy days indeed.