QUOTE: "Whatever is left over can be used for pressurizing other airplane functions."
There is nothing 'left over', the HPC (and IPC in the case of a 3 shaft engine) is simply robbed of air to supply cabin bleed and TAI. The R-R Trent 100 is different in this respect as it doesn't have any cabin bleed functions, only TAI, which is just one of the reasons it's a very fuel efficient engine.
Anyway, to answer the OP's original question, the primary function of the HPC is to supply hot, pressurised combustible air to the combustion chamber, not to augment the thrust supplied by the LP compressor. The small amount of thrust generated from the hot nozzle is a bonus, but never the less taken into account during the design process.