Your relationship with your investors and your board is both public and private (or at least, it's supposed to be). Yes, you are owed a private space to discuss difficult internal matters. But your investors are also responsible for getting you ready to be a public company, and that means holding you accountable when you've fallen below the threshold of what would be acceptable to public shareholders. As the letter notes, "accountable" by definition means in the open and out of your total control.
Mitch and Freada are doing Uber's shareholders a service (and therefore also doing Travis a service) by helping the company clean house before they're exposed to the harsher sunlight of being a public company. If you wouldn't have them as investors as a result, then you're not ready to be held accountable by others and aren't public CEO material.