Nothing is more frustrating than having everything freshened up and it still has starting trouble.
Your reference to a leaking float bowl maybe a clue. That particular engine has quite a bit of harmonic resonance. If the float seat assembly was worn from vibration it could be flooding some fuel into the carb throat. This excess raw fuel would require a few kicks when it's cold to un-flood the engine when it's cold. When it's hot the same amount of fuel within the engine cases vaporizes because of the temperatures creating too rich of a fuel air ratio for ignition.
I would change the needle and seat. Make certain to use the same style and number that is stamped on it. Some are designed for fuel pumps and have a smaller office and won't flow enough in a gravity supply like on a dirt bike. The float arms should be paralell to the gasket surface when you hold the carb upside down. Adjust it by bending the tab pressing on the end of the float needle. Also be certain there is no dirt in the pilot and choke passages from the inlet side of the carb to those devices.
A sure fire way to start a 250 2 stroke that's cold and has no problems is, pull the choke, keep the throttle closed, push the kick starter slowly about 5 or 6 strokes, get
it coming onto the compression stroke, let the starter go back to the top, then kick it like you're trying to break the foot peg off.
If a 2 stroke has cooked crank seals, most likely it will be a bit oily behind the flywheel or you will have a lot of blue smoke, not white smoke, from it sucking in the transmission oil on the PTO end of the crank.
Oh, and the large screw with a visable spring around it is for idle speed. The air screw, or pilot screw is a very small brass one quite close to where the air box joins the carb. Counter clockwise is more air or leaner. Clockwise is richer. It effects idle and low throttle characteristics.
Let us know if you solved the problem already.