When you are talking about bleed air, the only reference is from the "core" or gas generator part of the engine. Commonly this seems to be denoted as "the core" "N2" or "Ng".
The bleed air extraction can come from 1, 2, or 3 places along the compressor section, often referred to as "LP" - Low pressure, "IP" - Intermediate Pressure or "HP" - high pressure. Engines usually have a source of bleed air for their own internal use ( pushed through the turbine stages to keep them cooler). Also there is a means for systems required extraction, (anti-ice, packs, hydraulic reservoir head pressure).
As pointed out by the other posters, most air going through the core is used for cooling and flame control. As you pointed out, there is advanced research into flame patterns and burning all the fuel versus pushing it unburned out the back (think of old school jets with thick black smoke trails). However, the more air we take from the engine, the less we have to control flame propagation and cooling the turbine stages. If we take too much air, the flame will actually touch the burner can and the resulting effect will be much the same as taking a cutting torch to that metal.
When you accelerate a turbine engine, you first see an EGT increase as there is more fuel added, then as the rpm increases, you get more airflow and the egt temp decreases as there is more cooling air once the engine stabilizes at a set rpm. You can also see the effect of HP and LP valves opening and closing. The best way is on an engine that automatically switches, in the ground. It's a hot day outside, so you want as much air as possible through the packs. You can push up the power on one engine, watch the EGT rise, then settle back. However, if you keep pushing the power up, you'll see the EGT suddenly cool as the HP valves close and the LP valves open. Same reason on a cross-bleed start, most engines require a specific N2 to start the opposite engine. It is close to the limit of the HP valve. If you push the power too far, you'll have the HP valve closed then the LP valve opens and you may have more power on the engine (80% N2 vs 70% N2) but less air to actually start the engine.