While its good practice to add the 125%, and everyone does that in practice, I think the question is technically incorrect because it references part III of article 220 which has no such requirement for a continuous load. If the article 220 was going to reference another article requiring the continuous factor it would be in the beginning under the section title “Other Articles”.
In addition 230.31(A) Covers service conductor ampacity and also has no such requirement.
There are examples of other appliances that are treated as continuous loads for 'branch circuit' sizing but not for 'service conductor' sizing.
And using part III implies service conductor sizing.
Every step in the question has a code reference, except the continuous part.