I think that the phenomena is not stable under perturbation of the metric, i.e., a small perturbation can cause the minimizer to be smooth.

First, let me explain how it is not stable under perturbation of the "link."

If you have a smooth submanifold $\Gamma^{n-1} \hookrightarrow \mathbb{S}^n \hookrightarrow \mathbb{R}^{n+1}$ so that the cone over $\Gamma$, $C_1(\Gamma)=\{tx : x\in \Gamma,t\in [0,1]\}$, is area-minimizing, then there exist arbitrarily small perturbations $\Gamma_\epsilon \hookrightarrow \mathbb{S}^{n}$ of $\Gamma$ so that the solution to the Plateau problem (i.e., the area minimizer with boundary $\Gamma_\epsilon$) for $\Gamma_\epsilon$ is regular.

In general, this is a consequence of the work of Hardt and Simon, who showed that if $C(\Gamma)=\{tx : x\in\Gamma, t\geq 0\}$ is area minimizing (this is equivalent to $C_1(\Gamma)$ solving the Plateau problem for $\Gamma$), then $\mathbb{R}^{n+1}\setminus C(\Gamma)$ is foliated by smooth, area minimizing hypersurfaces which are asymptotic to $C(\Gamma)$ at infinity. For the Simons' cone you mentioned, this fact was proven earlier by Bombieri, De Giorgi, and Giusti as part of their proof that the Simons' cone is area minimizing.

Now, because these smooth surfaces foliate $\mathbb{R}^{n+1}\setminus C(\Gamma)$, and are asymptotic to $C(\Gamma)$, we can find one, say $\Sigma_\epsilon$, which intersects $\mathbb{S}^n$ in a (smooth) surface, say $\Gamma_\epsilon$ which is arbitrarily close to $\Gamma$. Because $\Sigma_\epsilon$ is area minimizing, $\Sigma_\epsilon\cap B_1(0)$ must be the solution to Plateau's problem. Moreover, it is the unique solution thanks to the fact that the $\Sigma_\epsilon$'s form a foliation (e.g., by the maximum principle or a calibration argument).

Now, via the above facts, I claim we can construct a small perturbation of the Euclidean metric so that $\Gamma$ bounds a unique area minimizing surface which is smooth. To do so, simply construct a diffeomorphism $\phi:\mathbb{R}^{n+1}\to\mathbb{R}^{n+1}$ so that $\phi(\Gamma_\epsilon)=\Gamma$. Clearly we can arrange that $\phi$ is close to the identity in whatever sense we want. Now, for the metric $\phi^*\delta$, the solution to the Platau problem for $\Gamma$ is unique and smooth, because it is $\phi(\Sigma_\epsilon)$.

I think that the phenomena is not stable under perturbation of the metric, i.e., a small perturbation can cause the minimizer to be smooth.

First, let me explain how it is not stable under perturbation of the "link."

If you have a smooth submanifold $\Gamma^{n-1} \hookrightarrow \mathbb{S}^n \hookrightarrow \mathbb{R}^{n+1}$ so that the cone over $\Gamma$, $C_1(\Gamma)=\{tx : x\in \Gamma,t\in [0,1]\}$, is area-minimizing, then there exist arbitrarily small perturbations $\Gamma_\epsilon \hookrightarrow \mathbb{S}^{n}$ of $\Gamma$ so that the solution to the Plateau problem (i.e., the area minimizer with boundary $\Gamma_\epsilon$) for $\Gamma_\epsilon$ is regular.

In general, this is a consequence of the work of Hardt and Simon, who showed that if $C(\Gamma)=\{tx : x\in\Gamma, t\geq 0\}$ is area minimizing (this is equivalent to $C_1(\Gamma)$ solving the Plateau problem for $\Gamma$), then $\mathbb{R}^{n+1}\setminus C(\Gamma)$ is foliated by smooth, area minimizing hypersurfaces which are asymptotic to $C(\Gamma)$ at infinity. For the Simons' cone you mentioned, this fact was proven earlier by Bombieri, De Giorgi, and Giusti as part of their proof that the Simons' cone is area minimizing.

Now, because these smooth surfaces foliate $\mathbb{R}^{n+1}\setminus C(\Gamma)$, and are asymptotic to $C(\Gamma)$, we can find one, say $\Sigma_\epsilon$, which intersects $\mathbb{S}^n$ in a (smooth) surface, say $\Gamma_\epsilon$ which is arbitrarily close to $\Gamma$. Because $\Sigma_\epsilon$ is area minimizing, $\Sigma_\epsilon\cap B_1(0)$ must be the solution to Plateau's problem. Moreover, it is the unique solution thanks to the fact that the $\Sigma_\epsilon$'s form a foliation (e.g., by the maximum principle or a calibration argument).

Now, via the above facts, I claim we can construct a small perturbation of the Euclidean metric so that $\Gamma$ bounds a unique area minimizing surface which is smooth. To do so, simply construct a diffeomorphism $\phi:\mathbb{R}^{n+1}\to\mathbb{R}^{n+1}$ so that $\phi(\Gamma_\epsilon)=\Gamma$. Clearly we can arrange that $\phi$ is close to the identity in whatever sense we want. Now, for the metric $\phi^*\delta$, the solution to the Platau problem for $\Gamma$ is unique and smooth, because it is $\phi(\Sigma_\epsilon)$.

EDIT: I realize that I should have referenced the following paper of N. Smale http://www.ams.org/mathscinet-getitem?mr=1243523 "Generic regularity of homologically area minimizing hypersurfaces in eight-dimensional manifolds." where he uses this sort of argument to prove that a $7$-dimensional minimizer in an $8$-manifold will be smooth for generic metrics. I should also point out (as Smale does in the article) that this is very much unresolved in higher dimensions. This has a lot to do with the poorly understood nature of the structure of the singular set of minimal hypersurfaces in higher dimensions.