The circuitry on the PCB, aside from the large rectifier, is responsible for the analog meter and LED indicators only.
Upon closer inspection it appears to have a relay to turn the output on/off, presumably upon sensing that the battery has reached fully charged voltage. Is this relay being energized appropriately when a discharged battery is connected to the charger, and then de-energizing the relay when no battery is connected? If yes, then this circuitry is doing what it should.
Otherwise, this charger is nothing more than a cheap and nasty unregulated unfiltered half-wave rectified power supply. The output from the transformer passes through a single rectifier diode, a thermal overload, and a current shunt.
More than likely you've got a bad connection somewhere, high resistance, so as soon as you put a load on it all the voltage drops across the bad connection.
It is also possible the bad connection is on the primary side, so check to see if the voltage to the primary taps on the transformer is proper.
Check for cracked solder joints, bad switch contacts, loose terminals. Use a DMM to determine where the voltage drop is occurring.
Do also check across the relay when it is energized, in case it has poor contacts.