Is there an origin in any of the schools of thought for the action of the Imam who prayed alone in his room after missing the congregational prayer, when the prayer was established due to his lateness, or is it an error contrary to Islamic law?

If the regular Imam is late in coming to the mosque, and the worshipers believe he will not attend or the prayer time is becoming short, then they should put forward one of them to lead them in prayer, and their prayer is valid. The Imam should not leave them if he arrives while they are praying.