Why was Moses chosen to lead the Jews out of Egypt?

Because God chose him to do it. God didn't need to take auditions for the role of "Jewish Redeemer." I mean, He didn't search for a candidate with the right qualities to pick as leader. From the beginning of Time, He had orchestrated the Egyptian Exile and Exodus, and had chosen the main actors.

God had selected and dispatched the unique soul of Moses to Earth as part of His original master plan. The Torah describes how Moses was picked for the job for our benefit, so we could learn what it takes to qualify as a Jewish leader.

We know that Moses was pre-selected for the position, because from the day he was born his home was filled with a spiritual glow. He was also born circumcised, another sign of innate holiness.