The whole concept was necessary for one of two reasons, and often for both.

The first reason was most nations could not produce either individually or enmasse turbochargers running on engine exhaust. Outside the US, who was the leader in this particular niche, no other nation made extensive use of exhaust driven turbochargers on aircraft.
The alternative was a mechanical supercharger run off the engine. As these grew in size their parasitic load grew with them. For very large systems it was more efficent to just have a dedicated engine running the supercharger compared to each engine running its own compressor.
The US also ran into this problem with the P-47 being the classic design case. Here you have an aircraft with such a large turbocharger supplying the engine that there is not enough exhaust to provide power to run it alone. So, the designers added a drive shaft making the system a turbosupercharger that combined the "something for nothing" exhaust turbocharger with the necessary make up by parasitic load of supercharging.

The Germans, Japanese, and Russians really had little choice but to go with supercharging. They lacked the metallurgy to build more than a handful of turbochargers. In the Japanese case they actually built one prototype fighter (a take off of the Ki 84 Hayate) using captured B-29 superchargers from shot down aircraft! This hardly bodes well for mass production.
Outside the Fw 190C the Germans pretty much stuck to the supercharger system. The Hs 130 was mentioned. I believe there was a Do 217 with a similar system that was tried also. Most of the time, using just the supercharger was sufficent for typical altitudes reached.
In any case, these systems had a practical limitation of about 45,000 feet in any case. Much above this piston engines just were not going to get enough air regardless of the system used. Going higher demanded jet engines.