Temperature control fails. Will a PID solve the problem?
just on/off with an offset

You probably mean an ON/OFF controller with hysteresis? The ON/OFF controller is always stable, meanwhile the PID is not. The PID controller has to be tuned to work properly. It can be very painful to properly tune it, even for an engineer that has a big theoretical knowledge on closed loop control. The ON/OFF control drawback compared to PID, is that ON/OFF control produces higher overshoots/undershoots as it is constantly oscillating around the setpoint value, but for rude regulation it is OK. Also not complicated, easy to setup.
If this kind of control can't provide a stable oscillating control value around setpoint, neither a PID would be able to control.