If we assume that we're observing at a random point in our history, we are equally likely to be observing at a point in the first 10%, second 10%, third 10%, and so on. That's what the assumption means. Which means there is a 10% chance that we're observing in the last 10%.

See https://en.wikipedia.org/wiki/Doomsday_argument for the most famous use of this line of argument. (It concludes that the human species is likely not long for this universe.)

What's stopping us from applying this math to other things? Like, every baby at the 1 minute mark of their life. Shouldn't that mean that ~10% of all babies born dies within a minute and a half?

We can take this to extremes. At the 1 minute mark we have a 99% change of being in the latter 99% of their life (and thus a 99% chance of dying 99 minutes later). It's amazing how many of them beat these odds.

It's an uninformative prior (https://en.wikipedia.org/wiki/Prior_probability#Uninformativ...). Meaning it's the best estimate you can make, if you don't have any additional information about the problem. Obviously we have a lot of additional information on human lifespans, which allows us to build a more accurate probability distribution.

Also see my comment above.

Nope, your example would be true if we didn't know how long babies lived, and we were observing a baby who was a minute and a half old.

There's something seriously squirrely about that math.

I really don't think thats true and I also don't think that your doomsday argument link relates to what you are claiming either.

The conclusion absolutely follows from the assumption. But there are two questions.

1. Is that a reasonable assumption to make? (In a surprisingly wide variety of cases, yes. But not always.)

2. How do we update the conclusion of the argument given a variety of other information that we have available to us? For example we have a great deal of information about the US system of government, current political state, potential threats, and so on.

He's saying that if you looked at the whole history of the US and divided that into 10 equal parts, then there would be a 10% chance of any random observation falling into the last part. Hence, all else equal, we have a 10% chance that we are looking at the US in the last 10% of it's history.

This would imply that there is also a 10% chance we are currently looking at the first 10% of the country's existence, which means you have confidently calculated the odds of the USA lasting at least 2350 years at 10% based on absolutely nothing. These probabilities are nonsense.