Why did the Shaykh mention only one opinion—that a woman covering her face is "the soundest of sayings"—even though most scholars say otherwise, and why did he not mention that this saying is based on his own opinion?

The expression "the soundest of sayings" means the saying that has the strongest evidence, not necessarily the saying of the majority of jurists. The preponderant opinion is that it is obligatory for a woman to cover her face, and this is what the Qur'an and Sunnah indicate, as well as the practice of believing women and the fatwas of contemporary scholars. The Permanent Committee, headed by Ibn Baz, answered that a woman's face is 'awrah (private part) according to the soundest opinion. In case of disagreement, one must refer to the Book and the Sunnah. Among the evidences for the obligation of covering the face are: Allah's saying: (O Prophet! Tell your wives and your daughters and the women of the believers to draw their cloaks all over their bodies), and His saying: (And tell the believing women to lower their gaze and guard their private parts and not expose their adornment except that which [necessarily] appears thereof). Also, Ibn 'Abbas's interpretation of the first verse indicates the obligation for women to cover their faces.