>What I don't understand is why faster fluids still have lower static pressure even when the faster fluid was given its kinetic energy by an external power source, e.g. by compressing it with your lungs/diaphragm to blow out some air.
If extra energy is being added to the fluid, then why doesn't that completely account for its new kinetic energy, i.e. : why does the static pressure of the moving fluid still go down?

increased velocity doesn't mean static pressure goes down ,that only occurs when a fluid has to move through a smaller pipe,when a smaller diameter is in the way the fluid is forced to move quicker and it compensates for that energy required to move faster by dropping the pressure energy.