How do we interpret the verse: "O you who have believed, when the believing women come to you as emigrants, then examine them," taking into account that whoever declares Islam is to be judged as such, unless a nullifier [of their Islam] appears?

What the questioner mentioned, that the rulings of this world are based on outward appearances, is correct, and the principle is not to scrutinize hearts. However, the reason for the Prophet, peace be upon him, testing women and not men, is that the Treaty of Hudaybiyyah included the return of anyone who came emigrating as a Muslim from Quraysh. So, Allah, Glorified and Exalted, excluded the matter of women with this verse, and ruled that a believing female emigrant should not be returned to the land of disbelief. Rather, she should observe an iddah (definition) (waiting period) and then be married, and her disbelieving husband should be given the dowry. And their testing was not a form of incapacitation, but rather it was based on the clauses of the pledge of allegiance for women, which are: that they should not associate anything with Allah, nor steal, nor commit unlawful sexual intercourse, nor kill their children, nor produce any falsehood, nor disobey him in what is right.