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The common core engine arrangement is why I'd have thought they'd have gone with that configuration from the start, using the aux stage literally in place of a turbocharger (sans intercooler ... though, technically, an intercooler could have been installed as well, just as with a turbo).Practically speaking, using the carburettor between compressor stages allowed the core engine to be, essentially, the same as used in a turbocharged installation. The downside is that charge cooling could only come using ADI, and not an aftercooler.
I believe it also shortened the length of the whole engin without that massive carburettor hanging off the back.
This is counter-intuitive given the intermediate carb installation requires a longer duct with more twists and turns leading from the aux stage to the carb than is the case for the aux stage directly into the engine stage.The change improved the altitude performance not because of the carburettor position, but because of the reduction in losses in the air flow for 1st stage to 2nd stage.
Why would intercooling reduce the power required to drive the second stage? The air-fuel mixture is not burning in the intercooler, it is only getting cooled. If the after cooler brings the charge to the same temperature before getting injected into the cylinders, then the power produced would be the same.
Now if the aftercooler doesn't quite get the charge cooled down to the same temperature as without the intercooler, then adding an intercooler while retaining the aftercooler would lower the charge temperature and possibly produce slightly more power, but it doesn't change the power required to drive the seconds stage in any way.
Inter-coolers are provided between successive stages of a multi-stage compressor to remove the heat of compression hence reduces the work of compression (power requirements). The work of compression (power requirements) is reduced by reducing the specific volume through cooling the air. Thus inter-cooling affects the overall efficiency of the machine.