Why did the early Jerusalem believers stay in Jerusalem instead of going home?
Early Jerusalem believers stayed because they expected the Messiah's Kingdom to arrive immediately. Since the King was coming, they saw no reason to return to Babylon, Rome, Spain, or North Africa. They remained in Jerusalem, living from the common fund as a stopgap until the Kingdom would be established.
Les explains that after Pentecost, many Jews believed Jesus was the Messiah and that the Kingdom was just over the horizon. Because the King was coming, they asked why they should go back to Babylon, Rome, Spain, or North Africa and live among pagan Gentiles. So they stayed in Jerusalem and lived out of the common kitty as a stopgap until the King would come and the Kingdom would be set up. But Israel did not respond, the King did not come, and the kitty ran out. They ended up poor, having sold their goods, unable to buy a ticket home, caught in Jerusalem. Les says God is just, and these people were caught in a legitimate cause because they had believed the Gospel of the Kingdom. God would take care of them until natural death through the agreement Paul made with Peter, James, and John to remember the poor saints in Jerusalem.