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I think @Joel is saying the following (this was too long for a comment, so I put it as an answer):

Given a conical symplectic resolution $X\mathrel{:=}T^*M \rightarrow X^\text{aff}$ whose $\mathbb{C}^*$-action contracts the fibres of $X$ with weight 1, the ring of global functions $R=\operatorname{Spec} (H^0(X,\mathcal{O}_X))$ has a set of homogenous generators $\{x_1,\dotsc,x_n\}$ with maximal weight 1. (**)

The last fact, together with the normality of the ring $R$, is precisely saying that $X^\text{aff}$ is a conical symplectic singularity with maximal weight 1, thus the Namikawa's theorem applies, stating that $X^\text{aff}$ must be a closure of a normal nilpotent orbit in a semisimple $\mathfrak{g}$. Then, by Fu_Symplectic Resolutions for Nilpotent Orbits (Theorem 0.1), we indeed have $X \cong T^*(G/P),$ for some parabolic subgroup $P$ of $G$ (and $\operatorname{Lie}(G)=\mathfrak{g}$). Moreover, $X^\text{aff}=\overline{O}_P\mathrel{:=}\operatorname{Im}(\mu),$ where $$\mu:T^*(G/P)\rightarrow \mathfrak{g}^*\cong \mathfrak{g}$$ is the moment map of the $G$-action (and $\mathfrak{g}^*\cong \mathfrak{g}$ is obtained via Killing form).

Though, what still worries me is that the argument above does not cover resolutions $$X=T^*(G/P) \rightarrow X^\text{aff},$$ where the induced map $X^\text{aff}\rightarrow \overline{\mathcal{O}_P}$ is not an isomorphism, but rather a finite cover. This is a general setup, and in some cases (e.g. when $G=\operatorname{SL}_n$), this map is indeed an isomorphism.

Having this in mind, perhaps the statement (**) about the weights of global functions is wrong?

I think @Joel is saying the following (this was too long for a comment, so I put it as an answer):

Given a conical symplectic resolution $X\mathrel{:=}T^*M \rightarrow X^\text{aff}$ whose $\mathbb{C}^*$-action contracts the fibres of $X$ with weight 1, the ring of global functions $R=H^0(X,\mathcal{O}_X)$ has a set of homogenous generators $\{x_1,\dotsc,x_n\}$ with maximal weight 1. (**)

The last fact, together with the normality of the ring $R$, is precisely saying that $X^\text{aff}$ is a conical symplectic singularity with maximal weight 1, thus the Namikawa's theorem applies, stating that $X^\text{aff}$ must be a closure of a normal nilpotent orbit in a semisimple $\mathfrak{g}$. Then, by Fu_Symplectic Resolutions for Nilpotent Orbits (Theorem 0.1), we indeed have $X \cong T^*(G/P),$ for some parabolic subgroup $P$ of $G$ (and $\operatorname{Lie}(G)=\mathfrak{g}$). Moreover, $X^\text{aff}=\overline{O}_P\mathrel{:=}\operatorname{Im}(\mu),$ where $$\mu:T^*(G/P)\rightarrow \mathfrak{g}^*\cong \mathfrak{g}$$ is the moment map of the $G$-action (and $\mathfrak{g}^*\cong \mathfrak{g}$ is obtained via Killing form).

Though, what still worries me is that the argument above does not cover resolutions $$X=T^*(G/P) \rightarrow X^\text{aff},$$ where the induced map $X^\text{aff}\rightarrow \overline{\mathcal{O}_P}$ is not an isomorphism, but rather a finite cover. This is a general setup, and in some cases (e.g. when $G=\operatorname{SL}_n$), this map is indeed an isomorphism.

Having this in mind, perhaps the statement (**) about the weights of global functions is wrong?

I think @Joel is saying the following (this was too long for a comment, so I put it as an answer):

Given a conical symplectic resolution $X:=T^*M \rightarrow X^{aff}$ whose $\mathbb{C}^*$-action contracts the fibres of $X$ with weight 1, the ring of global functions $R=Spec (H^0(X,\mathcal{O}_X))$ has a set of homogenous generators $\{x_1,\dots,x_n\}$ with maximal weight 1.** The last fact, together with the normality of the ring $R,$ is precisely saying that $X^{aff}$ is a conical symplectic singularity with maximal weight 1, thus the Namikawa's theorem applies, stating that $X^{aff}$ must be a closure of a normal nilpotent orbit in a semisimple $\mathfrak{g}$. Then, by Fu_Symplectic Resolutions for Nilpotent Orbits (Theorem 0.1), we indeed have $X \cong T^*(G/P),$ for some parabolic subgroup $P$ of $G$ (and $Lie(G)=\mathfrak{g}$). Moreover, $X^{aff}=\overline{O}_P:=Im(\mu),$ where $$\mu:T^*(G/P)\rightarrow \mathfrak{g}^*\cong \mathfrak{g}$$ is the moment map of the $G$-action (and $\mathfrak{g}^*\cong \mathfrak{g}$ is obtained via Killing form).

Though, what still worries me is that the argument above does not cover resolutions $$X=T^*(G/P) \rightarrow X^{aff},$$ where the induced map $X^{aff}\rightarrow \overline{\mathcal{O}_P}$ is not an isomorphism, but rather a finite cover. This is a general setup, and in some cases (e.g. when $G=SL_n$), this map is indeed an isomorphism.

Having this in mind, perhaps the statement ** about the weights of global functions is wrong?

I think @Joel is saying the following (this was too long for a comment, so I put it as an answer):

Given a conical symplectic resolution $X\mathrel{:=}T^*M \rightarrow X^\text{aff}$ whose $\mathbb{C}^*$-action contracts the fibres of $X$ with weight 1, the ring of global functions $R=\operatorname{Spec} (H^0(X,\mathcal{O}_X))$ has a set of homogenous generators $\{x_1,\dotsc,x_n\}$ with maximal weight 1. (**)

The last fact, together with the normality of the ring $R$, is precisely saying that $X^\text{aff}$ is a conical symplectic singularity with maximal weight 1, thus the Namikawa's theorem applies, stating that $X^\text{aff}$ must be a closure of a normal nilpotent orbit in a semisimple $\mathfrak{g}$. Then, by Fu_Symplectic Resolutions for Nilpotent Orbits (Theorem 0.1), we indeed have $X \cong T^*(G/P),$ for some parabolic subgroup $P$ of $G$ (and $\operatorname{Lie}(G)=\mathfrak{g}$). Moreover, $X^\text{aff}=\overline{O}_P\mathrel{:=}\operatorname{Im}(\mu),$ where $$\mu:T^*(G/P)\rightarrow \mathfrak{g}^*\cong \mathfrak{g}$$ is the moment map of the $G$-action (and $\mathfrak{g}^*\cong \mathfrak{g}$ is obtained via Killing form).

Though, what still worries me is that the argument above does not cover resolutions $$X=T^*(G/P) \rightarrow X^\text{aff},$$ where the induced map $X^\text{aff}\rightarrow \overline{\mathcal{O}_P}$ is not an isomorphism, but rather a finite cover. This is a general setup, and in some cases (e.g. when $G=\operatorname{SL}_n$), this map is indeed an isomorphism.

Having this in mind, perhaps the statement (**) about the weights of global functions is wrong?