They probably shouldn't use the old name of the person, as people can change their names for all sorts of reasons, and there's not really any good argument for refusing to use a person's updated name at their request.
However, pronouns are rather different, as there is an established usage of using 'he' for males and 'she' for females, i.e. in reference to the immutable quality of a person's sex. If you don't believe in the ideology of gender identity, it's more honest to keep using a person's sex-based pronouns, regardless of their preferences to the contrary.
However, pronouns are rather different, as there is an established usage of using 'he' for males and 'she' for females, i.e. in reference to the immutable quality of a person's sex. If you don't believe in the ideology of gender identity, it's more honest to keep using a person's sex-based pronouns, regardless of their preferences to the contrary.