elementary number theory - Understanding the proof of "$sqrt{2}$ is irrational" by contradiction.

I have some difficulties in understanding the proof of "$\sqrt{2}$is irrational" by contradiction. I am reading it in 10th class(in India) Mathematics book( available online, here )

This is the snapshot of it:

The proof starts with assuming that $\sqrt{2}$ can be written as a ratio of two integers and then that this fraction can be reduced to its lowest terms i.e. $\sqrt{2}=\dfrac ab$, where gcd(a,b)=1 . Then at last we reach at the contradiction that gcd$(a,b)\neq1$. Then they say that because of this contradiction $\sqrt{2}$ cannot be a rational number.

What I do not understand is that how the contradiction proves that $\sqrt{2}$ cannot be a rational number. The contradiction only proves that $\sqrt{2}$ cannot be written as the ratio of two coprime numbers. But can't we write $\sqrt{2}$ as the ratio of two non-coprime numbers?

Let us consider two statements, X and Y as:

X: $\sqrt{2}$ cannot be written as the ratio of two coprime numbers.

Y: $\sqrt{2}$ cannot be written as the ratio of two non-coprime numbers.

The contradiction proves only the statement X not the statement Y.

I guess that we can prove statement Y from X as:
Let us suppose that $\sqrt{2}$ can be written as the ratio of two non-coprime numbers, i.e. $\sqrt{2}=\dfrac RS$, where $R$ and $S$ are mutually non-coprime. But every rational number can be written as a fraction in lowest terms. So let's say $\dfrac RS$ in its lowest terms is $\dfrac rs$, but this means that $\sqrt{2}$ is also equal to $\dfrac rs$, where $r$ and $s$ are coprime. This eventually contradicts the statement X, hence by contradiction $\sqrt{2}$ cannot be written as the ratio of two non-coprime numbers, or the statement Y is true.

Question:

1. Did I prove the statement Y from X correctly ?

2. Why does the book directly mention "$\sqrt{2}$ is irrational" without justifying statement Y? Is the justification too trivial to be mentioned?

3. Is there any other way than mine(proof by contradiction) to deduce Y from X?

I only want to clarify these three doubts, nothing else.