I generally agree with your idea.I don't see the intergalactic current coming in as a single Birkeland current right to the surface or photosphere of the Sun. That would be one Mother of a current which we would not have missed. Rather, the intergalactic current feeds into the heliopause, which in turn discharges into the heliosphere, which turns into currents which feed into the corona. The corona has discharges which create currents that feed into the chromosphere. Finally the chromosphere discharges currents into the photosphere. (And I probably missed out some spheres here.)
During the solar anode / cathode debate on this forum I proposed NEITHER.
The sun is another type of unique, self organised, electrical phenomenon. IMO, as is a Double Layer.
I am suggesting the interior body of the sun acts as GROUND, with the other half of the circuit being the Interstellar medium outside the heliosphere.
The "drift current" (as described by Dr. Scott in his book The Electric Sky) ensues, and the sun happens.
I also don't think there is a single direct Birkeland current feeding the sun.
The solar interior is so massive that it can be one half of it's circuit. The term capacitor seems to fit.
I don't think there is a dominant current PINCH that supplies the sun either.
Although the concentration of the plasma within the heliosphere at the sun is similar to a pinch, I think it is a different phenomenon.
The solar system is a plasma sphere with a GIANT ROCK in the middle where all the plasma goes.