Is it permissible for a wife to pray alone at home while her husband leads the prayer there due to an excuse preventing him from going out, and does the husband's lack of proficiency in recitation justify this?

There is no harm for a woman to pray alone in her home. It is preferable for the spouses to pray in congregation together to attain the reward of congregational prayer. If the husband makes an error in reciting Al-Fatihah that alters its meaning, then it is not permissible to follow him in prayer. However, if his error does not alter the meaning but rather violates the recommended rules of Tajweed, then it is permissible to pray behind him.