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.
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).
I don't see why that arrangement would preclude use of an aftercooler, though. Use of a standard modular core engine would preclude it, yes, but the inter-supercharger carburetor placement seems perfectly compatible with aftercooling. (an intercooler in series with the aux stage and carb intake would be more compatible with the modular add-on arrangement though and technically COULD be of liquid cooled type and not air to air, and would probably be best located at the carb intake, minimizing bulk and length of glycol plumbing to only slightly more than the aftercooler arrangement does)
Mimicking the behavior of a turbocharger really seems like the best/most foolproof direction Allison could have taken and is probably why they discarded the fixed gear ratios in favor of a fluid coupling relatively early on. (even a single-speed fluid coupling would allow pretty useful performance, particularly with a full-neutral setting avoiding excessive oil/fluid heating at low altitudes) I wonder if examining the impeller size and RPM range of GE's turbos would have accelerated development more than building off Allison's own superchargers alone (or potentially outsourcing to Wright or P&W)
I know GE's supercharger designs were quite lacking in the 1920s and early 30s (leading to P&W and Wright investing in developing their own) but the added competition led GE improving their own hardware quite a bit from what I understand. (and even if the compressor design was inferior to what Allison was already working with, the mass flow, diameter, and operational RPM range would be useful for testing -including avoiding stall conditions between the two stages)
For that matter, running an auxiliary supercharger of identical construction to the integral stage, but driven totally independently (off some sort of supplemental powerplant or perhaps even electric motor) could have been used for initial trials and rather quickly written off that sort of pairing as unworkable, or established what range of operating conditions it was workable within and whether those were worthwhile. (without wasting time actually developing the 2nd stage's mechanical or fluid coupling mechanism)
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.
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.
Reduction in intake losses from the carb-less aux stage intake manifold seems to be the big gain here, possibly in addition to charge cooling between the two stages being more efficient than at the intake of the aux stage.