Why do companies do this, instead of "pay cut, take it or leave it?"

It's never phrased that boldly. People aren't "fired", their contracts are not renewed, or they are declared redundant because of the new kid with an advanced degree... usually just after they've finished training the new kid.

The most common move is to cut the highest paid section of the workforce after they acquire the service contract from a competitor. All major defense firms do this, which leads to problems maintaining institutional continuity.