I've never gotten into board level diagnosis so sorry, can't help there. However, there are some simple visual inspections you can perform.
First, if you're not comfortable with handling electrical stuff, don't go any further. Contact an electrician or Authorized AutoPilot Service Center.

Second, if you are ok with electic, turn the power off to the systems.
Third, remove both covers to get to the circuit board.
Forth, inspect the circuit board for any signs of discoloration, burns, or damage to the components on the board. Especially in the upper left quadrant, where the bridge rectifier and capacitors are located, look for damage, such as the capacitors swelling up (and indicator that they have leaked/failed).

Regarding the partial check system display, that's a new one on me. You may check the wire harness to ensure they are all plugged in correctly. The only display that involve "Purifier Off, Check ____" is Check Flow. If salt is at 0 ppm, it could very well be due to insufficient flow through the manifold assembly so it's not getting flow, and not allowing the trisensor to read the salinity.

Hope this helps,