In many cases, if the business is advancing toward a downsizing, they might choose not to make a counteroffer and instead use the departure to fill a planned staffing reduction. This can pose serious issues for the manager when a business chooses such a course when their manpower was not originally slated for downsizing.
I've seen lower appetite for counteroffers in the "middle range" of maturity. When a team has begun to reject cowboy culture and tribal knowledge, embracing high levels of testing, documentation and automation there may be reduced interest in keeping someone for skill and knowledge. I've seen this behavior diminish when organizations reach a higher maturity level and are making a more sizable investment in long-term growth of their employees through things like training.
When serving as a manager, there are many cases where one might step in to ask the business not to make a counteroffer, such as: if the employee is relocating due to their spouse, if the employee determines remote work is not a good fit for themselves, or if the employee is seen to be making positive career steps forward which could not be matched in their current work environment. Many managers are big enough people not to try to hold their employees back.
There are of course many places I've not worked, and many possible cases I've not seen, but the above is my experience.