You have a relay that bumps the voltage to the blower up but only if the signal gets to it. There is also usually a problem with the A/C control switch which is spring loaded and vibrates with the car so that the points often burn out if not fully engaged. It is poorly designed at best and the whole thing could be better served with a less decorative and more functional switch, such as one with detents to hold the switch in place once set.
I won't even mention the quality of off shore parts bought at the corner discount auto parts store where the focus is on maintaining a low price by selling only low quality parts.
You will probably have to check each for continuity and resistance.