It's exactly the other way around - if they dismiss him for a vague general reason, they're much less exposed to litigation than they would be if they falsely accused him of lying.

You are 100% correct here, which is how we can reasonably conclude that the accusations were not false.

If the accusations by the board are true, that doesn't explain why Brockman and a few of the senior researchers quit as a response to all of this.

Them leaving does not imply accusations are false. They may like him, they may dislike new boss regardless of accusations, they may dislike overall future direction. They may think they would be fired some times later regardless.