Rolls (and later everyone else) got around this problem by fitting a supercharger driven by a two-speed gearbox and/or (depending on model) a clutch mechanism to engage a second compressor stage. This was used with a conventional butterfly throttle which essentially controlled air inlet pressure, with the supercharger gears/stages kicking in when barometric switches said they wouldn't cause knocking.
RR's 2-stage superchargers were on the same shaft. Meaning that there was no option of de-clutching the 1st stage (nor the 2nd, closer to the engine), so both stages were turning at same RPM.
RR's concept was copied - too late - by Jumo for the 213E and F, as well as by DB for the 603L and 605L.
P&W did it differently on their 2-stage supercharged engines. The auxiliary S/C (1st stage) had 3 settings: neutral, low, high. In neutral, the ram air was not going through the aux S/C, but was routed directly to carb and the engine-stage S/C. In low setting, the aux impeller was running at lower speed, and ram air was going 1st through it and then through the carb and the engine-stage S/C. Similar with high setting, just this time the aux S/C was turning faster.
Engine-stage S/C was single speed, ie. no possibility to chagne the gears there, and was turning when engine was on.
Allison's 2-stage set-up (used eg. on P-63s and P-82s) was with the engine-stage S/C remaining the 1-speed unit, and the aux stage was run via hydraulic coupling and thus was with variable speed. Both stages were always on when engine was on.
- The Sydlowski throttle is interesting in that it did two things at the same time - while it acted like a butterfly throttle, it also imparted a lot of swirl to the air going into the centrifugal compressor. Since the compression ratio of a centrifugal compressor is a function of the net amount of tangential velocity added to the air passing through it, adding swirl will reduce the compression ratio rather than (as in the case of a butterfly throttle) choking off the air mass flow to reduce inlet pressure so that the outlet pressure is reduced.
- This essentially widens the operating range of the compressor where it can operate acceptably - probably (in combination with a pressure relief valve on the outlet to avoid over-boosting at low altitude) meaning you can get a single-speed system to work across the whole altitude range. This could be single or dual-stage, although initially single-stage would be a lot easier.
- The first US source I can find writing about this is July 1941, with an engine arriving for test (with support from M. Planiol) at Wright Field by December 1941.
French (Turbomeca - company of Mr. Planiol and Syzdlowsky) installed that peculiar system on their supercharger on the HS-12Y engine, that became the -45 (and later the -49 that sacrificed a bit of power down low to gain a bit more above 5 km). The HS-12Y-45 powered the D.520 fighters.
Yes, the improved ways of throttling would've been beneficial for the engines of the day. Install something like that, or the Polikovsky's device (compied by the Germans on the Jumo 213s from the captured Mikulin engines) would've produce a Merlin III with perhaps 1000 HP for take off instead of 880, or 1400-1450 HP down low at +12 psi boost instead of 1300.
I'm not sure that the Turbomeca system was ever operated on a 2-stage engine, or even on the 2-speed 1-stage S/Ced engine. Perhaps test mules?