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There is a good paper of Goresky, "Triangulation of Stratified Objects", that I think reasonably quickly implies Milnor's result and its generalization to non-isolated singularities. The result is that any Whitney-stratified set, and in particular any algebraic variety in $\mathbb{C}^n$, is supported on a smooth triangulation. I think that you just need that and the inverse function theorem.

As I meant to explain in the comments, this theorem is has sometimes been regarded as a "chore theorem"chore" theorem. You can look at what Goresky says: "Triangulation theorems for stratified objects have been obtained independently by Hendricks (unpublished), Johnson (unpublished), and Kato (in Japanese)". When Goresky wrote his paper, it was a messy question that did not have a well-defined status. Now the situation is a bit better and I think that this generalization of Milnor's result can be called settled. Sometimes a good author not only proves a chore theorem, but also cleans it up an elevates it to non-chore status. But a lot of chore theorems are never proven in a clean form or are never proven at all.

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There is a good paper of Goresky, "Triangulation of Stratified Objects", that I think reasonably quickly implies Milnor's result and its generalization to non-isolated singularities. The result is that any Whitney-stratified set, and in particular any algebraic variety in $\mathbb{C}^n$, is supported on a smooth triangulation. I think that you just need that and the inverse function theorem.

As I meant to explain in the comments, this theorem is a "chore theorem". You can look at what Goresky says: "Triangulation theorems for stratified objects have been obtained independently by Hendricks (unpublished), Johnson (unpublished), and Kato (in Japanese)". When Goresky wrote his paper, it was a messy question that did not have a well-defined status. Now the situation is a bit better and I think that this generalization of Milnor's result can be called settled. Sometimes a good author not only proves a chore theorem, but also cleans it up an elevates it to non-chore status. But a lot of chore theorems are never proven in a clean form or are never proven at all.