"the perception of an employee wanting to leave a company goes from 'terminable flight risk' to 'an asset worth fighting to keep' as soon as you get an offer from another company."
This is so true. An offer changes the dynamic of the negotiation totally. I think the psychology and economics goes like this: If there is no counter offer, then the company would pay just more for the monthly asset they will likely have any how.
However, if there is a direct offer they are competing with, it means there is on the balance an acute need to find and train a replacement - one whom might actually cost more than the current employee is paid. Also, the value of the employee is validated by his demand in the job market.
The emotions, which will be present but subdued in a professional setting, are also different. Without a counteroffer, the employee will just sound greedy. With a counter offer, the employee is actually generous by offering the current employer a chance to save in hiring and training costs (unless the new wage is ridiculously high).
Silly and strange world we live in.