(The above rendering comes from the RJPS translation, an adaptation of the NJPS translation. Before accounting for this rendering, I will analyze the plain sense of the אִישׁ term, by employing a situation-oriented construal as outlined in “Notes on Gender in Translation,” pp. 11–16.)
Prototypically, אִישׁ is used in sketching a situation schematically. The fixed expression כְּאִישׁ אֶחָד describes the manner of action in an event, likening it to what would be expected if the situation had only one participant. Here, the expression is applied to the actions of an individual, Amasa. The expectation of informativeness implies that the analogy means that Amasa’s convincing the entire Judahite army was (in modern terms) a piece of cake—as easy as convincing a single person. I.e., there were no holdouts or objectors.
As for rendering into English, NJPS ‘as one man’ is inaccurate. According to Merriam-Webster’s dictionary, that English expression means “unanimously.” Such an adverb could apply only to the actions of the group. The revised rendering instead expresses in modern idiom the meaning that was arrived at above.