I don't think that's an "often" interpretation, but it is a potential conclusion to draw from it. Yes, you're correct that "someone has to be early, and maybe we just happen to be one" is entirely possible. But that is then something extra you have to include and defend with your theory of interstellar civilization spread.

To put it another way, the theory that most experts would accept right now suggests we should be in the middle of the distribution. If evidence shows we AREN'T, then either the theory is wrong, or you have to defend that "Yep, it's a good theory; we're just unusual."

A more general answer to your question is because it's a principal of astrophysical reasoning ever since the Copernican Revolution. Tradition (and religion) taught that the Earth had a privileged position in the universe. Centuries of subsequent scientific investigation showed that not to be the case for virtually every attribute measured, and so that has become a strong principle that guides scientific investigation in general. It's so rarely going to be true that claiming it is true for a particular theory is very suspicious (and more likely means there's something wrong with your theory).