How was the ground strap to the rectifier ?
if there is no ground the rectifier can not supply power to recharge the battery.
You test it by checking the purple wire to ground, engine off ignition on it should read battery voltage, if not there is a bad connection / broken wire.
Next start the engine and measure the AC voltage on the white wires, between them and from each one to ground.
You should get something better than 26V AC between them and 1/2 that between each one & ground.
You should also see 13 to 15 V DC between the purple wire & ground.
If you do not get 26V AC then pull off the flywheel and check for a broken wire where the wires go into the coils and make sure you have a good clean contact between the support legs and the crank case.
Also check the magnets, if some are missing you will get a reduced output.
If you can not get 26V AC then you can not get 12V DC and that is the minimum you need to recharge the battery.
As for the discharge, disconnect the battery overnight and see what it is like the next day if the battery goes flat when not connected then it is faulty.
If the battery does not self discharge then the easiest way to track the problem is to pull the connections in a dark shed at night and note which ones spark, none should.
The most common suspect is the rectifier, one diode goes open circuit then it tries to turn the alternator in to a very low power electric motor
