First of all, what carb do you have?
Yes the engine is supposed to run with the choke fully off (or open).
If you have an n.t. carb, the choke will be a simple, old style plate that opens and closes at the inlet of the carb. If you have to close the choke, it's telling you that you are getting too much air (making it run too lean). When you close the choke on an n.t. carb you are restricting the amount of air that the carb sucks in. The choke plate actually blocks the carb inlet.
less air means that it will run rich for cold start ups.
if it's an n.t., there is no air-fuel adjustment screw...just an idle screw.
If you have a cns carb. then the choke on this carb works backwards from the n.t. style carb. On the cns carb, when you choke it, you are actually opening the choke (which is just a brass shaft that slides up & down on the side of the carb inlet.) when you open the choke on a cns carb, it allows more gas to be sucked into the carb to make it run richer for cold start ups. more gas added to the air flow will make it run rich for easier cold start ups.
if it's a cns carb, then yes, there will be an air-fuel adjustment screw that you will have to play with.
so an n.t. carb...you close the choke to choke it when cold starting, and then open the choke all the way once it's warmed up. it should run excellent with the choke open.
on the cns carb...you open the choke to choke it when cold starting and then close the choke all the way once it's warmed up. It should run excellent with the choke closed.
but seriously, 9 times out of 10, if you have to leave the choke 1/2 way on, then you have an air leak somewhere.
check for air leaks at the carb with a can of starting fluid. Just spray a tiny bit around the carb intake gasket and the carb to intake manifold seal with the engine running.
If the engine revs up when you spray either of those areas, you found your air leak. also make sure that the top cap on the carb is screwed on straight, all the way and it's hand tight.
I know that you said that you checked already, but if you use starting fluid, the engine will rev up wherever there's an air leak.
also, if this is an n.t. carb, remove the air filter housing and look into the carb, Make sure that the choke plate is actually going all the way open and all the way closed when you use the lever.
If it's a cns carb,remove the air filter housing and look to the right side (i think it's on the right) of the carb opening. Operate the choke, and you should see a brass shaft move up & down every time you use the choke. you want that shaft open to choke it, and closed when it's warmed up. Make sure it goes fully open and fully closed when you use the cable operated lever.