The noise suppressor is a capacitor, usually like a 1 or 2 micro farad. They were used on wiper motors, blower motors, sometimes off the alternator or off the coil. RustyPile is right, don't need the suppressor usually, can just bypass it. Do you have a decent meter? After you make sure you have a good ground, not 100% sure, but I thought there are two separate sources for power also. High speed got power off the red wire and the other speeds got the power through the orange wire, both are on the plug at the fan relay. The purple wire is supposed to connect to one side of the suppressor and the other side of the suppressor connects to the blower motor. There is supposed to be a ground wire that comes down and connects to the screw through the suppressor bracket...in your second picture, what does that black wire that comes from above down to the suppressor connect to, can't see in the picture if it is ground or what it is.
This is what wiring should look like...
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If I knew how to photoshop, would have got rid of the question marks on electricity to fan, lol.