It seems like in SVM the best way to find out 'how to pass' is to find the shortest path from your point in the feature space to the separating hyperplane. The length of that vector in each axis explains how much you need to change to cross the threshold.

How about you run 20 vector machines ? Each only identifies how likely one reason is (using distance from separating plane as a "how sure is this SVM" measure). Besides, SVM do not have a particularly good reputation due to the need for good kernels, which requires either massive computing power (just have 10 million kernels) or people with very good domain knowledge. And if you need people with very good domain knowledge, why bother with AI ?

AI is hard and people who know both a lot about statistics, AI and programming AND a lot about, say insurance claims, well good look finding them. People with good knowledge about insurance claims and good knowledge of excel, they're, well, not a dime a dozen, but certainly much easier to find.