Generally with a cold engine, you will need some choke applied to keep the engine running. As the engine warms, the choke can gradually be opened. I don't recall you mentioning what style of choke your carb has, manual or electric and whether it is functional. That's one thing.
For the idle mixture screws, the more turns out they are, the richer the mixture is at idle. Your engine is unable to idle right now, so that's kind of moot. A good starting point with the mixture screws is 2.5 turns out from gentle seating. Set them there and leave them alone for now. Both screws should always be an equal number of turns out.
You also have a curb idle speed screw. You should examine this screw to see if perhaps it has been backed off to the point that it is not applying any tension to the accelerator linkage.
Fuel should not be draining out of your fuel filter once the engine dies or is shut off. The only way your engine is able to run is at higher revs, so its burning fuel at a faster rate than you'd see at a true idle. Even so, a good pump will be able to stay ahead of even wide open throttle demand. Either the fuel pump is faulty or the carb is dumping excess fuel down the intake faster than the pump can replenish it.
The fuel filter drainage points squarely at the fuel pump. No other component would be responsible for this condition. Where is the fuel draining to? Back into the tank, or into the crankcase? I know you attempted a bench test and it seemed to pass, but I just don't know how much, if any faith you should put into those results. The real world indicators are pointing towards pump failure.