What external information? All the species are identical. Think of them as all the possible ways humanity could have gone, if a butterfly had flapped its wings. How is this not exactly the issue involved in a confidence interval? You are clinging to the 95% confidence without recognizing that being the 10 billionth changes that confidence irreparably, even if you don't know how it changes it. You may only maintain that confidence if you never use the information that you are the ten billionth. That is the extra information.
But that's not how confidence intervals work. Where is your idealized set, all who conclude their species won't outlive 200 million, and only 5% are wrong? That is the set you have not produced.