It shouldn't work "fine" with the choke pulloff disconnected, because the choke pulloff doubles as the vacuum brake for the secondary air door. Disconnect the hose from the choke pulloff at the carburetor. Now depress the choke pulloff into the diaphragm and put your finger over the hose where you disconnected the it at the carburetor base and let the choke pulloff go where you depressed it. If it stays put and releases when you take your finger off the hose, it's good. If it releases before you remove your finger, it's bad.
