In the sentence 'She suggested that they go to the cinema' there is no way of telling from the sentence in isolation whether it means that the speaker gave advice on attending a moving picture show, or whether the speaker believed that some people were already in the habit of enjoying cinematographic entertainment. The same is true in relation to an individual when the past tense is used, as in 'She suggested that he went to the cinema.' We depend on context to tell us which meaning is intended.
Now, when we come to the third person singular in the present tense, I understand that American English distinguishes between the mandative subjunctive ('She suggested that he go to the cinema') and the indicative ('She suggested that he goes to the cinema') to express the two meanings. Why does American English insist on an inflectional distinction in the third person singular here when it is obliged to rely on context elsewhere?