The pump operates when the pressure is below certain point, IIRC 140 bar. This is why when the car has sat overnight, the pump runs for 30-40 seconds before both the red and amber lights go out. Then, accumulator becomes critical – the stored pressure should be sufficient for 3-4 successive operations of the brakes
Regarding to two circuits — two of you are say the same thing – loss of pressure in the front means no brakes at all but I am still not convinced.
If this were true, then, how can the following observation regarding a condition where there is virtually zero pressure in the front be explained:
- bleeding Teves III front brakes with the two person method - nothing out of the ordinary - one person pumps the pedal, the other operates the bleeder valve.
However, if the two person method is slightly modified and the ignition key is turned to ACC II position, the electric pump can be heard operating when both bleeder valves (front and rear) are loosened…brake fluid flows from both bleeder valves.
In any case, this academic discussion can and should continue. However, the OP needs to be advised to replace and accumulator asap and properly bleed the brakes.