"Thus, when subtracted, all the '9's will cancel out exactly thus giving 9."

You cannot just subtract "all" the '9', since the number of '9's is not finite! This is Math, you have to prove it! Not just saying "all the '9's will cancel out exactly thus giving 9". I would say your argument is "ill formed".

Beside, let's take a finite sequence x = 0.999. Let's multiply by 10: 9.99. Now 10x - x = 9.99 - 0.999 = 8,991. This is valid for every finite sequence of type 0.999... which I've already wrote. So why, when we extend the reasoning to a infinite sequence, should this fact change? Again, this is Math world, you have to prove what you write, not just write random words like you did.

"which when compared to both the author's and my argument seems incredibly ill-formed."

If for "ill formed" you mean wrong, prove it! If you mean something like "complex, not elegant", than the funny fact is:

"The (elementary) proof, an exercise given by Stillwell (1994, p. 42), is a direct formalization of the intuitive fact that, if one draws 0.9, 0.99, 0.999, etc. on the number line there is no room left for placing a number between them and 1."

From https://en.wikipedia.org/wiki/0.999... under "Elementary proof".

That said my demonstration may be wrong, but still as an "elementary" proof, in wiki they used a demonstration that exactly "relies on it being absurd that there is no number between 0.999999... and 1"