Except the leaders of a competitor will almost always take a big exit and let the private company be bought, because fuck you, got mine. You only need to convince a few people to take a payday, and most people are not running businesses out of some strongly held ideology of public market competition, but rather because they want to be rich.

This is rarely difficult when the buying company is large enough. You just need to offer a good multiplier for the current price of the private company. Very few private companies will turn down a generous offer, especially when the option is to compete directly with the larger company.