Sounds to me like correctable problems...as in "problems, plural".
Keyster carb rebuild kits are good product. Frankly, however, I prefer the Dratv version, it has everything, including new main & pilot jets.
Best guesses...
A small vacuum leak, although the simple fact that the pilot airbleed screw has a noticeable effect might mean otherwise. Partial blockage of the emulsion tube wall orifices; those are smaller than the pilot jet orifice and can be a MoFo to clean completely, even with a thin wire. Might have low fuel level inside the float bowl; the pilot & main jets sit at slightly different heights above the bowl floor. That the curb idle (throttle stop) screw isn't doing anything is a red flag; i.e. something is assembled incorrectly. But, that should be an easy thing to identify & correct.
When rebuilding a carburetor, I like to set the initial curb idle adjustment by eye, while the carb is "on the bench". Look inside the venturi and turn the screw clockwise until you just the throttle slide just move a little above the completely seated position. That will at least show you where the adjustment range begins/ends. If the idle is too slow, which it should be, you'll know that a slight bit of clockwise rotation will (should) be all that's needed to get positive adjustment. I have the sense that you don't really know if the screw is moving the slide, or not, at this early stage.
FYI, main jets rarely clog up, unlike pilot jets. And I cannot recall seeing a vintage original jet needle that was worn, or oxidized (brass is remarkably stable metal) so you can reinstall those original pieces and if they work right, just leave `em.