On a Thinkpad R40, the mainboard pretty much comprises most of the components of the system. As far as the traditional responsibilities of the power supply, this is split up between the mainboard and the power adapter. The power adapter's job is to convert the power from AC to DC at a specific voltage. Once the power is fed into the machine, it gets converted to right voltages the different components of the computer needs.
The small explanation is yes, a mainboard replacement replaces the components that handle the job of the power supply.
I'm not sure exactly what might be causing your current problem and as you stated is hard to duplicate. Chances are though, if it was a power adapter problem, then the battery should kick in to compensate for the lack of power. Does the computer turn immediately after it shuts down?
Hardware Support Specialist