Any laptop should run with just the charger plugged in - no battery.
However, removing the battery WHILE the charger is plugged in might have caused other problems.
Unfortunately, laptop schematics are not readily available, if at all.
So, tracking down a bad part would be hunt and peck if you could recognise parts associated with the power circuitry.
IF there is a seperate power board, you may be able to replace that with positive results, BUT maybe not.
Otherwise, a MB replacement might work, but may not be economically prudent.