He suggested she should leave referred to an action that (at the point of time when he suggested) should happen in the future. "Should" has to be here because it determines the modus of the verb leave (here: she should go - it would be better if she went). Another possibility could have been "He suggested she left" this refers to an action (at that point of time) that has happened.
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I used to believe that the following is correct:
"He suggested (that) she (should) leave."