I think that it is significant that the Wolfen recipe calls for the potassium sufite solution to be at 55C. This increases the solubility of the p-aminophenol and helps the formation of the phenolate when the hydroxide is added. As more phenolate is formed more pap will dissolve. I would assume that the reaction takes place in a closed vessel to prevent oxidation.

BTW, there is a recipe in the Focal Encyclopedia of Photography that starts with p-aminophenol rather than the hydrochloride.