Let $\omega_{st}=\frac{i}{2}\sum_jdz_j\wedge d\bar{z}_j$ denote the standard 2-form on $\mathbb{C}^{n+1}$. The form $\tilde{\omega}=\frac{1}{||z||^2}\omega_{st}$ is invariant under the $\mathbb{C}^*$-action, therefore we should be able to push it down to $\mathbb{P}^n$ and obtain the Fubini-Study form. Moreover, if I understood well we can write an object on $\mathbb{P}^n$ in homogeneous coordinates if it is invariant under the $\mathbb{C}^*$-action.
I do not understand why in homogeneous coordinates the Fubini-Study form writes $$\omega_{FS}([z_0:...:z_n])=\frac{i}{2}\sum_j\frac{dz_j\wedge d\bar{z}_j}{||z||^2} -\frac{i}{2}\sum_{j,k}\frac{\bar{z}_jz_kdz_j\wedge d\bar{z}_k}{||z||^4} $$ and not simply $$\omega_{FS}([z_0:...:z_n])=\frac{i}{2}\sum_j\frac{dz_j\wedge d\bar{z}_j}{||z||^2} $$
Being invariant under the "global" $\Bbb C^*$ action as you're thinking of it is not sufficient; you need to allow a different action point by point. If you take any two local holomorphic sections of the $\Bbb C$-bundle $\mathscr O_{\Bbb P^n}(-1)$ and pull back $\omega$, you want to get the same $(1,1)$-form. (Think, for example, of what happens on the overlap of two standard holomorphic charts for $\Bbb P^n$.) Thus, you want not $\partial\bar\partial \|z\|^2$, but $\partial\bar\partial \log \|z\|^2$, which is in fact the Kähler form. Note that if $\lambda\colon U\to\Bbb C-\{0\}$ is holomorphic, then $\partial\bar\partial \log(\|\lambda z\|^2) = \partial\bar\partial \log|\lambda|^2 + \partial\bar\partial \log\|z\|^2 = \partial\bar\partial \log\|z\|^2$, as desired.