It is not "public health authorities". It is not a message from the government or any organization in a position of power. Instead, it is an open letter from health professionals.

They express themselves both as medical experts and as human beings, so having a political stance is no problem, as long as they make it clear which part of them is speaking. I think it is not clear enough in their letter, but in my opinion, there is absolutely nothing wrong with their message.

They're not expressing their support for the protests as a personal or political question separate from their medical expertise. The letter says that "white supremacy is a lethal public health issue", racism is "the paramount public health problem", and the protests are "vital to the national public health". This is beyond simply not making it clear enough; they're deliberately invoking their medical expertise as a source of authority for their political judgments.

The problem is not that they took a political stance. If a group of doctors wants to come out in support of Flat Earth, so be it. The problem is they potentially compromise the health of the public at the expense of their political message. Again, nothing was stopping them from acknowledging there is a risk, but that it is worth it.