Why do the Sunnis follow the four Imams, even though they did not live during the time of the Prophet, peace and blessings be upon him, and they do not follow the Household of the Prophet?

The heirs of the Prophet, peace and blessings be upon him, are the bearers of knowledge from among the Companions and those who followed them in goodness, whether they were from the Ahl al-Bayt (Household of the Prophet) or not. The Ahl al-Sunnah wal-Jama'ah (People of the Sunnah and the Community) follow all of them, and among them are many from the Ahl al-Bayt such as Ali and his sons, Al-Hasan and Al-Husayn, Abdullah ibn Abbas, Ali ibn Al-Husayn, Muhammad, and Ja'far ibn Muhammad, and other scholars like the four Imams, Al-Awza'i, Al-Bukhari, and Muslim. The Ahl al-Sunnah wal-Jama'ah follow the Prophet, peace and blessings be upon him, and those who followed him in belief and action, because their methodology is unified in creed, conduct, honoring the Sunnah, and adhering to evidence. The Imams became famous because their students preserved and recorded their schools of thought, which led to generations transmitting them.