Dumb question, but how does your flywheel look? Any evidence that it got rubbed by the coil? Is the yellow coating still visible on the magnets?
At least on this side of the pond, the solenoid “kit” with the filter, fuel line and solenoid is actually cheaper than the solenoid itself.
Anytime I have has a bad solenoid, it’s caused the saw to be pig rich, that’s the factory default. Even though you don’t see a spark, did you try a little squirt of mix down the carb bore?
I am fairly certain that the coil won’t fire without a solenoid connected, because many guys experimenting with swapping parts for performance gaind would have to leave the solenoid connected in the carb box when they swapped in non-mT carbs to play.
You can find used non-MT parts and convert it over to standard tune if you can’t take playing with it anymore.