Your basic error is that the turbine doesn't compress intake air.
The turbine extracts mechanical energy from the exhaust and delivers it by turning a shaft. In one case the shaft drives the crank, in the other the shaft drives a compressor. How does the turbine know the difference?
In F1 engines the turbine is connected to a system that both charges and compounds (by electrical connection). It is reckoned they are approaching 50% efficiency so I get the impression they have a fairly good idea about what they are doing.