Even if it's really a bad argument, the badness is far from obvious - just look at the Wikipedia page.

The fact that people are willing to believe something doesn't make it not obviously wrong. It just means they are, for whatever reason, blind to it's obvious wrongness.

For an example of why it fails in real-world terms consider the problem of coming up with the reference class. Humans? Great Apes? Apes? Mammals? Verterbrates? Earth-origin Living Organisms? Each produce a different prediction for the doomsday scenario, but a lot of plausible extinction paths for humans would at least take the rest of the apes with us.

For an example of why it fails the moment we have other evidence, consider Bob. Bob is 40 years old. He believes the doomsday argument. Someone points a gun at Bob, and threatens to kill him if he doesn't give up his wallet. Bob reasons "There's only a 0.001% chance that I'm in the last 0.001% of my life; so the danger of me dying in the next two hours is miniscule!". Is Bob right?

Now suppose that Sean has just turned 21, 3 months ago. Just become an adult. He concludes, from the doomsday argument, that as he's been an adult for 3 months, he has a 95% of stopping being an adult within 60 months, 5 years. So, he's going to die within 5 years?

A snap judgment cannot lead you from Zeno's paradox to discovering calculus.

No, but a snap judgement can lead you to correctly conclude that if each time you halve the distance you halve the time you're going to have a finite amount of time to cross the line, even if you have an infinite amount of instants.