Is it permissible for someone other than the muezzin to put forward a worshipper to lead the prayer in the absence of the Imam, given that the muezzin is present, and knowing that the muezzin is responsible for that?

The regular Imam of the mosque has the primary right to lead the prayer as long as he is present, because he holds authority in his mosque. If the regular Imam is absent, then the most rightful person to lead the people in prayer is the one among them who recites the Qur'an best. The muezzin or the mosque congregation may present someone to lead them in prayer in the absence of the regular Imam, provided that the affairs of the mosque remain orderly and no dispute arises, and the one who recites the Qur'an best is presented.