This may be true according to certain research or conditions. But it was not what was being taught in some of the text books of the time or in industrial practice(air compressors for ventilation or industrial processes). Mercedes had used a 2 stage compressor in their 1939 Grand Prix car (both stages were roots superchargers).
Superchargers were not well understood in the 1930s as the fuel would not support high boost. In the US the R-2600 is often said to be the first Wright engine to use a Wright designed supercharger. Others claim it was one of later R-1820 engines. Wright was building superchargers (very low pressure ones) from the late 20s on, but they were GE designs and in fact GE was often supplying at least some of the components which were often built by Allison under subcontract.
Obviously what a lot of designers 'thought' could change very quickly. Allison, P&W and Wright all figured out that GE superchargers weren't very good at about the same time (1-2 years?)
Also superchargers have sort of a sweet spot as far as efficiency goes. On a compressor map
View attachment 841951
We can see that peak efficiency of a modern, well designed compressor can vary from 78% down to the high 50s depending on flow with the worst being at the extremes.
At least one text book from 1943 (a few years out of date so as not to give away any secrets) was using 65% efficiency in most of their examples on inter coolers and temperature rise in aux superchargers with altitude. Being American there was little about 2 stage mechanical superchargers and it was assumed there would be an intercooler to reduce the intake air temperature before it entered the 2nd stage.
Turbo for a car is rather simple. Supercharger for an airplane is harder in that the mass airflow (pounds per minute?) is also affected by the density of the air and efficiency at cruise may not be peak efficiency at max power.
A Merlin III supercharger was maxing out at about a 3 to 1 pressure ratio and it was the best in the world in the late 30s for a variety of reasons (bad intake for one) so even after Hooker worked on the supercharger for the Merlin XX/45 when he wanted a 5-6 to 1 pressure ratio for the Merlin 60 he had to go to two stages. But the small Merlin (and Allison) need a lot of supercharging to move the air needed to make high power.
The problem with an inefficient supercharger is that not only does it take more power to drive the supercharger (compress the air to degree you want) but that the excess power is converted to heat over and above the heat created by simply compressing the air.
And the heat goes all the way through the engine. If you increase the heat by 100 degrees in the intake manifold you increase the peak temperature in the cylinder by 100 degrees and the exhaust temperature goes up 100 degrees.