Is it a sin for a trembling young man to lead people in prayer, and is his unsuitability for leading prayers considered a crime that necessitates abandoning the mosque in which he was raised?

Memorizing the Quran and making the Adhan are great acts of worship. Your leading people in prayer, when appointed to do so, is not a sin; rather, it is an act of excellence. Interrupting the recitation does not invalidate the prayer, so long as it does not affect the Al-Fatihah, and does not necessitate repeating the prayer. Fear and trembling when leading the prayer for the first time are natural and do not justify impermissible absence. There is no need for sadness or distress, as you have not committed any transgression. There is no blame upon you for leaving that mosque and going to another, for all of them are houses of Allah.