I will say I am not a commercial pilot, just a trainee, however my understanding is as follows.
The ACM speed is influenced by the amount of air passing over its turbine.
The air after passing the pack control valve can be sent one of two ways:
i) To the mix manifold without cooling
ii) Over the ACM for cooling.
It is the pack control valve that controlls the pack usage of bleed air. This will therefore increase bleed usage if the pack is put in high flow. It is the air mix valve that is controlled by the temperature controller - this routes the air over the ACM or direct to the mix manifold depending on the temp required. So temperature control appears to have no direct control over bleed air usage, just the path it follows.
Please feel free to correct me!!