Let us attempt to address the "silliness" using a very concrete construction. Imagine a supra-intelligent alien species that is the last surviving intelligence anywhere in the observable universe. A statistician from the species has access to all knowledge of every intelligent population that has come before. Specifically, they know the distribution N(n), where N is the number of populations that gave birth to n members before it went extinct.

Now, let us further stipulate that this statistician uses a random process to select one individual from all those populations, where each individual was equally likely to be the one selected. It seems to me this is precisely the situation at issue in the Carter reasoning. Now, the question is, if we look at the species that this individual was chosen from, it turns out they were individual number m from that population. The question is now, what is the probability that their species ever gave birth to more than 10*m individuals?

The Carter argument claims that probability is 10%. Hence, it must follow that if we repeat this experiment a billion times, in only a billion/10 times will n > 10*m. Unfortunately, that is a false conclusion, because the actual answer depends on the distribution N(n), no assumptions about which have been specified in the usual setup of the problem (essentially because nothing is known about that distribution). For example, if we take a simple example where N(n) corresponds to a 50% chance of n=10¹¹, and a 50% chance of n=10¹⁵, then any individual with m < 10¹¹ does indeed have a 50% chance of belonging to either population (as in Grand Marquis' scenario, made more concrete here). Ergo, for that N(m) distribution, when m=10¹⁰ (as it does for us), it is false that there is a 10% chance that n > 10¹¹-- the chance is actually 50% for that m value.

Now, it could certainly be argued that in the distribution I am talking about, it is extremely improbable that m=10¹⁰ will be chosen. Nevertheless, in the spirit of "someone has to win the lottery", the key point is you cannot ask those people to make a standard probability argument-- they are in a special place. So in that is that actual N(n) in our universe, then we know we have m=10¹⁰, so we have no way at all of knowing that we are not special-- if that is really the N(n), then the m=10¹⁰ individuals are making exactly the Carter argument, and getting exactly the wrong answer.

The way I sum this up is, you cannot make any probability argument that invokes both the number m and the average n in the same argument-- because if you don't know N(n), you will be reaching a false conclusion if you do that. You can use the average n-- you can say that before you choose m, the chances are that m will not be too different from the average n (but that requires knowing what the average value of n is). Also, if the statistician did not know N(n), and was selecting m value to suss it out, that's also fine-- the first m chosen begins to give the statistician a sense of what the average n is. But nevertheless, when all is said and done, when you bin all the m=10¹⁰ individuals together, which is the bin that we are in, you find that 50% of them live in a species that will vastly outlive 10*m.

This binning proves that the Carter argument is simply wrong probability, even if you make the very same assumptions it does. Any group that uses their own m value in their calculations of expectation values is specifying something about themselves that invalidates any claim they would otherwise have of being "generic". This is quite a common error in probability that comes up in a lot of puzzles, some on this forum.