Is the imam's repeating the prayer, based on the worshippers' alerting him to its invalidity because he forgot one prostration and then completed it, a sound action Islamically?

You were obligated to continue your prayer. Your return to complete the prostration of the third rak'ah is valid because you returned directly before reaching the prostration position of the next rak'ah. Your prayer is valid, and you must perform the prostration of forgetfulness (sujud al-sahw) after the tasleem. You erred by interrupting the prayer and restarting it, as interrupting an obligatory prayer without a valid reason is unlawful.