@IosifPinelis Yes, you are right, the positive dependence indeed stays for the 0-1 case, for every $n$. But as $n$ grows large, this dependence appears to be vanishing, Thus, we may say that for large $n$ if we delete a random 1 from an iid 0-1 sequence, then the leftover remains "almost" iid. I wonder, if there is any case where this vanishing does not happen, even as $n$ grows large. Or could it be true that any elimination rule leaves a leftover sequence which approaches an iid one as $n$ grows? Of course, one would have to precisely define what "approaches" means. Also, how fast is it?

@IosifPinelis The intuitive argument at the end of your answer does not seem to work for the 0-1 case, because the maximum is almost always the largest possible value (1). The only exception is when all terms are 0, but if the entries take value 1, say, with probability 1/2, and $n$ is large, then this happens with exponentially small probability. We may say that let's start with a conditionally iid sequence, where the condition is that not all terms are 0. Is it true that leaving out a random 1, the leftover will not be iid even in this case?

Nice example! Could the argument be carried over to the example mentioned in the last paragraph of the original question?