Here's a simpler way to see what is going on. Number all the humans that will ever live in order of their birth from 1 to N. The odds of your birth order number (call it B) being in the range of 1..M<N is M/N. The larger N is, the less likely it is that your B is as small as it is. So the odds of N being large (relative to B) are small.

Another way to look at it: half of the humans who ever live will have fewer people born after them than before them. Your a priori odds of being such a person are exactly 1/2.