Oh, at least in the post Soviet tradition of oral exams, you are given plenty of time to think and prepare your argument before you start an actual conversation with the examining professor. And if you need to think more during the exam to answer a tricky question, the professor just shifts to talk to another student who is ready to present their case for the time being. So it is less of a public speaking and more of a conversation where you can take a pause to think through an argument, or a proof, or a problem.