Why doesn't the company announce that they are going to cut about so and so many people in the next this and that timeframe? A lot of folks would go willingly anyway. (As this happens in other countries.)

Managers can pick before the announcement and then HR can do the actual if not enough people left willingly.

And there's always a chance of wrongful termination lawsuits...

... the real reason this is done this way is because that's the culture of these companies, mostly due to US culture (and labor laws are a synergic manifestation of this).