Way to assume your audience are morons.
It's not that hard with "20th century" which I'm conditioned to map to "last century" immediately, but when someone did something in the 17th century I first think "Oh like 1750-ish" and then I realize I'm a century off, and it disrupts my reading flow.
"The 1800s" typically refers to the years 1800-1809. "The 1900s" refers to 1900-1909. It is by no means unambiguous, but this is the common and plain meaning, and indeed these folks may confuse many more people by attempting in this manner to refer to 1800-1899 or such.
If we say "In the nineteen hundreds..." we mean 1900-1910. If we said "In the ninteen fifties" we would not be implying a 50-year period, would we? If we said "In the nineteen nineties" are we implying a 90-year period? Do you see how your logic immediately breaks down? Is it mere accident that "the ninteen tens" is a ten-year period of time? Yes it is...
Your assertions and these patterns of speech are just bizarre. I don't know anyone who says, or writes, "the Eighteen Noughts" at all. It is not a thing for scholars of history, for sure.
For a while it has been debated how we'll refer to 2000-2009. Because in different centuries, those initial ten years gained unique names. Some people want to call it "The Aughts" or "The Noughts" but I don't really hear people referring to "The Two Thousands" yet, at all; it's largely "The 21st Century".
Is there really much benefit in framing it this way?
I see the phrasing decision as raising several questions for the author:
1) Who is your audience?
2) What are your goals for the communication?
I could imagine either phrasing choice making sense depending on the answers.
Also, this isn't strictly accurate: the 19th century excludes 1800 but includes 1900.