Is the fact that Islam does not consider the face a private part (ʿawrah) in prayer, and that a woman prays with her face uncovered, evidence that the niqab is not obligatory and that a woman’s face is not ʿawrah?

It is not correct to use the uncovering of a woman's face in prayer as an argument that the face is not 'awrah (intimate part), because the command in the Quran to adorn oneself and beautify oneself for prayer differs from covering the 'awrah. A worshiper may be commanded to cover what is not 'awrah, such as a man's shoulder. A woman may cover in prayer what she is permitted to expose outside of it, such as her hair in front of maharim (unmarriageable kin), and yet expose in prayer what must be covered from non-mahram men, such as her face, hands, and feet, according to the majority of scholars.

Prayer has its own specific rulings, and what a worshiper covers in prayer is not necessarily 'awrah, and vice versa. Covering the 'awrah in prayer differs from covering it for viewing, and there is no connection between the two. It has been established by text and consensus that a woman in her home is not required to wear a jilbab, and she can pray with her face, hands, and feet uncovered. She is not required to cover her face, hands, and feet in prayer by agreement of the Muslims.