Is living in Madinah — which is 15 minutes away from the village — considered a shortcoming towards one's family, given the inclination to live there for the betterment of life?

The Prophet, peace and blessings be upon him, forbade neglecting the rights of parents, saying: "The parent is the middle gate of Paradise, so if you wish, lose that gate or preserve it." Therefore, it is not permissible for you to undertake anything that would cause hardship to your mother without her permission, especially if she has no one else to care for her but you. However, if it does not cause hardship and does not lead to neglecting your duty towards her, then there is no blame upon you.