"something very special about Earth's position that makes it far more likely than we initially calculate" - makes
what far more likely ? Spawning life? If so, then that special factor, whatever it is, would simply be an example of a great filter behind us. I'm not saying that we can't be the first ones, I'm just noting that us being the first implies that we're alone in the galaxy.
To make that point perhaps more clearly, let's say that before and including us, our galaxy has had X planets like Earth. Whatever that X is (1 or 1000 or billions), the total number of such planets that our galaxy will ever have is less than 2X. That doesn't rely on any poorly understood theories, that doesn't rely on any aspects of how life or intelligence might form - the lifecycle of stars is well understood, the age of universe is well understood, and the future has less new stars and planets than the past; we're currently past the midpoint of star/planet forming and that's that, most of the planets that will ever exist in our galaxy either exist right now (most of them for longer than Earth) or are gone already.
So whatever the "dice" of a suitable planet existing and creating intelligent life are - if we're the first ones (and I'm not making a point whether we are or aren't, whether that's likely or not, that "if" is an assumption), if X rolls of dice resulted in just a single success - then the remaining less than X rolls simply won't suddenly roll a hundred more successes; if the total number of spawned civilizations until us is 1, then that means that the total number of spawned civilizations until end of the universe would be 1 or 2 - and if it's 2, then it would be a hit-by-a-lightning-while-typing-this level of coincidence is that second one happened to arise at the same insignificantly tiny moment in time that constitutes the existence of homo sapiens until now, just some tiny 100 000 years after us instead of some 10 000 000 or much more years after us.
OR, possibly, there are and will be much more civilizations than that - but then some of them have been long before us.