The sigmoid will make effectively very hard for the network to produce values close to 1, because it would require a pre activation value close to infinity. Would this be a good behavior in your application?

Platt scaling does not have any guarantee and in fact it is easy to construct examples where it fails. On the other hand, conformal prediction methods, under very weak assumptions, on the multiclass problem of the question would give you a set of labels that is guaranteed to contain the true label with a specified probability.

The only approach that gives valid uncertainty quantification is conformal prediction, a quick google search should result in a good number of tutorials.

The training of decision trees is independent to monotonic transformations of each coordinate of the input features, this is not true for neural networks.

By law, US agencies have to disclose their use of AI. You can find a very long list by agencies here: https://www.ai.gov/ai-use-case-inventories/