First, the best way would be to get a schematic.
If you can see two solder "puddles" close to the battery input, the two may belong to one lead while a third "puddle" to the other DC input lead.
Radios often have voltage regulators after the battery input terminal (connector). Find the regulator pinout and test the regulated voltage after it. Measure the DC current from the battery to see if it is excessive (a short) or too low. Connecting the battery may only charge a capacitor but the radio may then not get the current with a proper voltage.
Battery connector may be a "switching" type, which disconnects the DC power from AC/DC converter when you plug the battery connector in it. Those switches as well as "volume" regulator switches often fail.
I once repaired a nice AM/FM radio by replacing a 12 V Zener which regulated the voltage for RF stages while the audio section was OK.
Working without a schematic requires experience, and still often fails.