Nice toy!
I still have my O1 stashed away somewhere in the basement, got it in the early '80s.
The video display on the O1 only shows 24 rows by 52 characters, so it is possible that there is nothing wrong with the video output, just a "regular monitor" is not able to synch correctly to the non-standard video output.
(The video buffer is 32x128, the 24x52 "window" on the screen can be moved around the buffer with keystrokes.)
I did expand the video circuit to output the standard display (24x80?) on an external monitor, it did work, though the image quality was not good, it was flickering, if I remember correctly.
There is/was a technical manual floating around the web "o1techm.pdf", 590 pages, 18MB, you should find it, it describes everything in details.
One of the problems was with the disk drives, Osborne decided to make his own drive electronics, and it was rather flaky. I remember getting up in the middle of the night trying to retrieve a file from a floppy when the HW was cold, the drive just failed after a while when in regular use.
I started to re-engineer the floppy interface for a standard floppy drive. I completed the electronics, but probably it needed some change in the firmware as well. I never completed that, I got a Mac SE-30 instead. The firmware listing is/was also around the net. It was not the source code listing, but the compiled firmware listing.
I did find an extra mobo in a surplus store on the way back, I did all the experimentation on that one.
Peter
EDIT:
Extracted and enclosing the video specs for the O1.
The horizontal sync is 64usec, the vertical is 16.6msec.
See if in the set up of your monitor you can select sync timings and if you can get close to these values.
You can also check with a scope if the video output sync signals are correct from the O1.