What's wrong in this method of solving a difference equation?

There's no particular solution of the form $y_n = an$, since, assuming $a$ is constant, you found that $a$ must satisfy $a=1/(1-n)$, contrary to the assumption that $a$ is constant.

Note that $a=1/(1-n)$ is not constant so there is no particular solution of the form $y_n=an$. On the other hand there is one of the form $y_n=b$ with $b=-1$ (which does not depend on $n$).

I think where you really went wrong is this:

You wrote that, if $y_n = an$, then $a = \frac{1}{1 - n}$. But the assumption $y_n = an = \frac{n}{1-n}$ is false to begin with, so the fact that you derived something from a false assumption means nothing.

Now why is the assumption false?

Well¹, according to your assumption $y_n = an$ we have $$y_n = \frac{n}{1-n}$$ right? So plug that into the original equation. Does it work?

$$\frac{(n+1)}{1-(n+1)} = 2 \frac{n}{1-n} + 1$$

Remember, this is implied by your assumption. But this only holds for $n = -1$.

In other words, it won't work for any other $n$ than $-1$... neither $-2$, nor $0$, nor $1$, etc...

So your assumption that $y_n=an$ holds for all $n$ contradicts itself, hence it cannot be true.

_{¹ Someone else contended that you derived this incorrectly too, but that's a math error separate from what I'm trying to show, which is the mistake in your reasoning. I just assume you did the math right and show where the logic went wrong.}