You think that the problem is in the motor; that the motor cannot supply more than 33 mA. This isn't correct, the problem is that your 12V fan is not able to function at such a low voltage and thus does not draw much current. Remember: current isn't "pushed" by the source, it's drawn by the load. If you were to short out the rectified output and measure the short circuit current with a multimeter, I guarantee you will see several amps flowing trough (possibly blowing the multimeter fuse, destroying the diodes and/or overheating the motor coils in the process).
A standard silicon rectifier diode will drop 0.7 V when conducting. As a rectifier bridge has two such diodes always in series with the load, the diodes will steal 1.4 V from whatever meager voltage your motor puts out.
You can rectify this by using schottky diodes, which have a much lower voltage drop (about 0.2 V), and using a lower RPM/V (KV in radio control terms) motor. For example, a 140 RPM/V (KV) brushless gimbal motor should put out 21.5 V AC at 3000 RPM, which can then be rectified and dropped down to a nice stable 12V or 5V with a buck converter.
If you want to stick with your motor for whatever reason, and you have the necessary programming skills, you could also write a custom firmware for a commercially produced ESC (RC brushless motor driver) to act as a synchronous rectifier instead. The ESC MOSFETs will drop next to no voltage when conducting and they are already arranged in a full bridge configuration, granting much better efficiency than any diode. You could even boost the motor voltage while rectifying by pulse width modulating the MOSFETs, doing exactly what the ESC normally does but in reverse.