Is the Imam obliged to let someone else lead the prayer if he suffers from hoarseness or a scratchy throat, fearing that his voice might give out, or is it permissible for him to lead with the intention of appointing a successor and returning to the position of a congregant to teach people this Sunnah?

You are not required to appoint a successor if you fall ill during the Maghrib prayer. If your voice breaks or it becomes difficult for you to recite aloud, you may recite silently, as both audible and silent recitation are sunnah (recommended) and not obligatory.

The imam is also not obligated to appoint a successor if he is unable to recite anything other than Al-Fatihah.

However, if you appoint someone else to lead the people in prayer while you are ill and fear being unable to recite, this is good. It is preferable for the imam of the locality to appoint a successor if he is ill.