What supervised machine learning model can be used to generate a scorecard-like result?
It depends what you mean by "can be used": any regression algorithm can be used, the question is how reliably it would perform. You can compare different algorithms experimentally (if you have a dataset). 



[Updated after question edited]

In general the way to use ML with this kind of setting is to train a classification model based only on the categorical features. Depending on the type of algorithm, the combination of features might not always be a weighted sum, and the result label may or may not be based on a cutoff point. In order to have a cutoff point (thus a numerical prediction), the method must be a soft classification method. Alternatively a regression model could be trained for predicting the numerical value.

So that leaves you with many options: 


soft classification: linear/logistic regression, Naive Bayes, ...
regression: linear/logistic regression, SVM, decision trees, ... 


Note: technically the probability doesn't represent "how far from the cutoff point", it represents the probability of the instance being positive (p=1).