I thought isolation valves were only for a tank or tank valve failure, such as a tank neck o-ring, or a burst disk - is this wrong? Wouldn't you in the case of a free flow merely turn off the knob at the first stage, ie just turn off the single valve? You wouldn't have to touch the isolation valve in this case, which is normally open anyway. Still, as you said, you use it to isolate the gas supply in each tank. If you burst a disk or a neck o-ring, you lose that tank's gas regardless.
I don't dive doubles but plan to one day, I had a different understanding of the role of the isolation valve.