A rectifier regulator is two things combined into one unit.
The rectifier part converts the alternating current produced by the stator coils into direct current for battery charging. The regulator controls the voltage.
There is also a signal output for the tachometer.
Small engines with low powered stators usually have an unregulated rectifier. Larger engines with more powerful stator coils usually have a regulated one to avoid overcharging the battery.
The fact that your tachometer does not work may be because the rectifier regulator is defective. Or, of course, the tach could be defective!
You should check the volts, Beg, borrow or steal a voltmeter.
Or even buy an inexpensive multimeter ( no home, car or boat owner should be without one! ) No need to spend a King's ransom on an expensive one.
Measure the volts at the battery. A good, fully charged battery, after resting for say 12 hours, should read somewhere around 12.7 volts.
Watch what happens when you start your engine.
If all is well the volts will fall when you crank the engine but should then recover and slowly rise to around 14 volts, or a little more, as the battery recharges. The lower the state of charge of the battery the longer that will take but if you start with a well charged battery it should take no longer than a few minutes.
If you do not see this recovery and subsequent rise you have a problem to investigate.
You may have a defective rectifier or there may be a bad connection.
Check all connections.
If possible repeat the voltage reading between ground and the red wire from the rectifier to confirm that it is the rectifier at fault rather than a bad connection
There are two things that will blow a rectifier. One is connecting the battery with the polarity reversed. Even briefly ... instant death!
The other is disconnecting the battery while the engine is running. You might get away with it at idle speed when the current is low but otherwise it is likely to cause a voltage surge which will blow the diodes. A loose connection could cause this.
Always keep your battery fully charged. If you leave a battery even partly discharged you will shorten its effective life.