Hey Armchairengineer (I like that handle!)...you are partially correct here.
As boost increases, so also does the overall compression that air undergoes as it passes thru the engine/turbocharger system.
HOWEVER it’s only the mechanical compression (actually the mechanical expansion…) taking place in the cylinder that results in mechanical energy showing up at the crank.
Increasing compression/expansion external to the cylinder (i.e. thru the turbocharger) affects combustion air density, and therefore fuel burn, maximum combustion temperature and ultimately power output. But it does not add to the compression ratio of an engine from a thermodynamic cycle standpoint.
There is however one exception to the above: Turbocompounding. Here the turbocharger is geared to the crankshaft thru an overrunning clutch and a bunch of gears. Under load, a portion of the mechanical energy extracted from the exhaust gases by the turbo is fed thru the clutch and gears back to the crank. This extracted energy adds slightly to the engine’s output. Therefore with turbocompounding a portion of the expansion occurring thru the turbo exhaust turbine DOES effectively add to the overall expansion ratio thru the engine.
Turbocompounding was successfully employed in some large WWII era aircraft engines. Then Mack (and probably others….) experimented with it briefly on diesels in the mid-70’s. But apparently they learned the small gain power/efficiency apparently are not worth the added cost and complexity. ...FB