Does the sin fall away from the men of the village if the Imam prays in congregation with his young son, who is 10 years old, knowing that congregational prayer is a communal obligation (fard kifayah)?

The correct view according to the Shafi'i school is that congregational prayer is a communal obligation (fard kifaya). Their evidence is that this position was explicitly stated by Imam Shafi'i himself and is supported by authentic hadiths.

The obligation is fulfilled if the symbol of the congregation becomes apparent, even if only a small group prays in the village. It is a condition that those by whom the obligation is fulfilled must be free and adult.

Based on this, the obligation is not lifted from the people of the village by the prayer of the Imam and his non-adult son, and they incur sin according to those who hold that congregational prayer is a communal obligation.