Actually lots of CFM users do extract fan discharge air. It is not used by the pneumatic system, but blown across the heat exchangers to cool the core air for a/c use.
I know more about the JT
than about the CFM, even though I have not flown one in many years. On that engine there were utility airbleeds at the 6th, 8th and 13th stages of the compressor section.
BTW the turbine section is not used for airbleed largely because it contains carbon monoxide, being exhaust from the burner section.
The only plane I recall using 6th stage was the 727. My ignorance there. But all users tapped primarily the 8th stage for a/c and pressurization. If needed, the augmentation valves would open and add 13th stage air. This had to be pre-cooled (with fan discharge air) because what we needed was more pressure or more mass flow, but not higher temperature.
Anti-icing and deicing require hotter air and may favor the 13th stage some.
On the CFM-powered Airbus the arrangement will be similar. Most air of an intermediate stage of compression, augmented as needed by highest stage air.
Happiness is not seeing another trite Ste. Maarten photo all week long.