I am simply not qualified to say anything insightful about the math point you make. The presumptuous philosopher doesn’t bother me, but that may just be scope insensitivity talking.
On the Great Filter, I agree that believing in the Filter and science discovering things like life is created easily (including complex life), Sol-like suns are common, and Earth-like planets occur often with Sol-like suns makes it seem likely that the Filter is in front of us. But it doesn’t say when. The Great Filter is consistent with The Crystal Spheres, which the anthropic Doomsday argument just isn’t.