Why can't we use the law of cosines to prove Fermat's Last Theorem?

How do you get to the conclusion that $ab\vert s$? I honestly can't see it. The way I see it you have:

$$\dfrac{r}{s}=\dfrac{(a+b)^2-c^2}{2ab}. $$

Now $(a+b)^2-c^2$ is even. You can check this case by case, when $a,b$ are odd then $c$ has to be even and so forth. So at least one of them is even but by your assumption maximal one is even and therefore $(a+b)^2-c^2$ is even. Therefore

$$ \dfrac{r}{s}=\dfrac{\dfrac{(a+b)^2-c^2}{2}}{ab}. $$

But there is no apparent (at least not to me) reason why this shouldn't reduce further. If it does your argument breaks down at this point.

Here is an actual counter example: of course I can't give an example of $a,b,c$ with $a^n+b^n=c^n$ but your argument that $ab\vert s$ only uses that $a,b,c$ are coprime. So let $a=13, b=15$ and $c=22$ than you have that $a,b,c$ are relatively prime and furthermore:

$$\dfrac{r}{s}=\dfrac{(a+b)^2-c^2}{2ab}=\dfrac{10}{13}, $$

therefore $s\neq ab=195$.