Duty of executor to present will for probate.) (a) Within 30 days after a person acquires knowledge that he is named as executor of the will of a deceased person, he shall either institute a proceeding to have the will admitted to probate in the court of the proper county or declare his refusal to act as executor. If he fails to do so, except for good cause shown, the court on its motion or on the petition of any interested person may deny him the right to act as executor and letters of office may be issued by the court as if the person so named were disqualified to act as executor.
     (b) When 30 days have elapsed since the death of the testator and no petition has been filed to admit his will to probate, the court may proceed to probate the will without the filing of a petition therefor, unless it appears to the court that probate thereof is unnecessary and failure to probate it will not prejudice the rights of any interested person. Such notice of the hearing on the admission of the will to probate shall be given to the persons in interest as the court directs.