elair, the choke pulled means less air going in, which effectively enriches the air mixture. This means either too much air (vacuum leak)...
Or since these carbs adjustment screws are actually fuel screws...opening them allows more fuel in...removing the main screw means you are supplying more fuel.
So its very possible from your description there is not enough fuel available during running conditions thus you must modify the supply of air/fuel to get it to run.
So...I'd look at the float level, needle and seat, fuel flow to the float bowl, etc.
If its only getting a little fuel in there, choking it or removing the screw but no choke levels out the air/fuel ratio.
Another possibility is a major vacuum leak. Like a warped or loose intake. And one last thought, how is the compression?
Keep us posted, this sounds like one of those weird and rare problems we can file away the answer for whenever the next time it pops up!