What is wrong with using t-SNE for predictions? If I have a dataset with hundreds of samples and thousands of features, and t-SNE does a good job of separating classes compared to others classifiers, I don't understand why I can't rerun the algorithm with an additional test sample and predict its class based on KNN. Even if the model is non-parametric, can't the robustness of the method in generating coherent clusters time after time (different seeds or omitting some of the training samples) be used as an argument for using this method empirically?

t-SNE gives no function for embedding out-of-sample data in the low-dimensional space. Consequently, all of the usual machine learning notions about out-of-sample performance are out.
If you use a different dimension reduction approach, such as UMAP or PCA, and then develop a functioning model based on that reduced dimensionality, that’s fine.
Addressing your exact question about KNN, if you use KNN after a dimension reduction, you’re still using KNN as the classifier, not an unsupervised method.
