The positions of leadership that the hadith refers to is those of the Imam (both of a nation and of the male/mixed congregational prayer), judges, and chief commanders of the Muslim army and those who carry out judicial punishments. However, even in regards the position of a judge, there were some scholars who permitted woman to be judges. [Fathul Bari, Tuhfa al Muhtaj, Fathul Muβin, Ihya Ulum al Din].
The reasons for the prohibition is more out of practical considerations rather than ability, as in many cases woman are often more capable and efficient than their male counterparts. The ruling also takes into account womenβs nature, which is usually much more compassionate and merciful than men, and while these are positive qualities in themselves, are not always what is needed in matters of leadership and command.
As for other positions of authority, such as scholars and teachers, heads and executives, managers, representatives, and advisors, even at the highest levels, there is no shariah prohibition to this, and women have equal rights to such positions, as well as being entitled to command equal respect and rights, including salary.
Sayyidna Umar (may Allah be pleased with him) appointed Samrah bint Nuhaik as the chief supervisor of the marketplace, and gave her powers to carry out her role. It is said that βShe would patrol the market while enjoining good and forbidding evil. She would discipline people with a whip that she had with her.β [al Istiβab fi Asma al Ashab]