This makes me want to play alpha centauri.

No alien civilizations have substantially colonized our solar system or systems nearby. Thus among the billion trillion stars in our past universe, none has reached the level of technology and growth that we may soon reach. This one data point implies that a Great Filter stands between ordinary dead matter and advanced exploding lasting life. And the big question is: How far along this filter are we?

Where is everybody?

We've only been broadcasting for about a century out of ~5 billion years of our star-system's existence (0.000002% of the total span), and only seriously listening for extraterrestrial signals for a couple decades. That's a pretty narrow window, and the odds that some other civilization's window lines up with ours may be pretty small.

And not only can the Drake Equation not tell us anything about the values of any of these variable, the equation can't even assure us that these are the only variables of interest.

If you convert the Drake equation to a lower bound and treat it as an expected value it is almost true as a matter of pure logic & probability.

300 years ago, radio waves were unknown. It wasn't until 1864 that mathmaticians even posited their existance. Given that, what will we know 300 years from now that we have no imagination to encompass?

This is what I had in mind when I was suggesting that the Drake Equation, even with all the variable unknown, isn't true as a matter of "pure logic." (I don't think we need to invoke anything as unlikely as a "singularity", though.)

300 years ago, radio waves were unknown. It wasn't until 1864 that mathmaticians even posited their existance. Given that, what will we know 300 years from now that we have no imagination to encompass?

I sympathize greatly with the notion that intelligent alien life will probably be a lot less like us than we are likely to think.

However, I don't think it's fair to say that prior to the discovery of radio waves, radio waves were beyond the imaginative capacity of humans. And they certainly didn't require ascension to a transhumanist state in order to comprehend them. Humans from before the discovery of radio waves would have been perfectly capable of communicating with and understanding humans from after the discovery of radio waves. I think generalizing from our past ignorance is good justification for undermining assumptions that there won't be radical discoveries and paradigm shifts around the bend, but that's a far-cry from a technological singularity. For one thing, you're already positing that you know what the new paradigm shift is going to be like, which kind of goes against the whole gist of the example which is to demonstrate how little we have known in the past or how little we have been capable of foretelling what will develop in the future.