To what extent is it permissible for professors to encourage their students to engage in simple exercise after prayer to improve physical fitness, given the concern that it might become a habit?

There is no objection to setting a specific time for permissible sports, and there is no objection to it being after prayers, as long as it is not believed to be part of the prayer. It is also permissible to play sports in the mosque if it entails a general benefit for Muslims, as evidenced by the Hadith (definition) of Aisha (may Allah be pleased with her) regarding the Abyssinians playing in the mosque. Al-Hafiz Ibn Hajar clarified that playing with spears is not mere play, but rather training and preparation for the enemy, and that the mosque is designated for the benefit of Muslims.