The metal shaft should never come out, it is press fitted in the motor when they are built and is intended to be permanent. This generally means the drive unit seized to the shaft by calcium deposits and the friction and rotation worked it loose over an extended time. The key here will be does the shaft still fit tight, if so you can put a drop of superglue on the shaft and tap it in with a hammer until it is fully seated, none of the shaft should be above the well, you have to work fast before the glue cures. This is not an option if the seat is badly wallowed out, then only a new motor is a viable repair. If you get the shaft seated, the disk pushes down the shaft the blue rubber faces down to the motor, the slotted face of the black plastic disk faces up to the impeller and sits in the cup.
The loose shaft causes that, if the propeller leans to the side and touches the side wall it binds up. Tap the shaft in well, it needs to be all the way in.