Is it true that a husband helping with housework is sunnah, not just generosity?

Yes, helping the family at home was the Prophet's ﷺ own practice, not merely an extra kindness. His wife was asked what he did at home, and she answered: “He was always in the service of (helping) his family, and when the time for prayer came, he went out to pray” (Sahih al-Bukhari 676). A husband who helps does not lose his manliness or his leadership; rather, he revives the sunnah and lightens his wife's load. Start with one fixed task that you take over completely, not just ‘helping’ when asked. Children who see their father helping learn that housework belongs to everyone. See responsibilities of husband and wife.