Brake System light revisited
So the distro block switch piston should be self centering.
If it's new I would expect it to work properly.
Assuming you got everything connected correctly...
You could try assisting in centering the piston by bleeding the front or the rear which will move the piston one way or the other.
When the e brake is engaged that also lights up the brake warning light, so be sure that is not the case, during your testing.
The fact that it goes out with brake application could be from air in the system allowing the piston to move, but does not explane why the piston will not self center.
You could remove the switch from the distro block, and physically check that the piston is centered.
IIRC there should not be any brake fluid where the switch is.