How can we reconcile what is narrated about some of the Salaf, such as Omar ibn Abd al-Aziz not performing ghusl for janabah or a wet dream since assuming the caliphate, with the Sunnah of the Prophet, peace and blessings be upon him, in showing affection to his wives and fulfilling their rights, despite his bearing greater burdens?

This narration about Fatimah bint Abd al-Malik concerning Umar ibn Abd al-Aziz has an unsound chain of transmission, as it includes Sulayman ibn Humayd al-Muzani, who is unknown in status (majhul al-hal).

Even if it were authentic, his wife consented and relinquished her right, and this is not considered a mistake or a violation of the Sunnah (definition) if it was with her full consent.

As for why he did not act as the Prophet, peace be upon him, did, it is because the Prophet's state is the most perfect of states. However, not everyone can emulate him in every single thing. Sometimes, different Sunnahs might conflict, and what is more important is prioritized.

Umar ibn Abd al-Aziz inherited a caliphate where injustices were widespread. He could not combine the duties of a husband with the duties of the caliphate, so he prioritized what was more important, which were the rights of the subjects, any shortcomings in which would be difficult to rectify, unlike the rights of a wife, for which an apology can be made.

He is similar to some great scholars like Al-Nawawi and Ibn Taymiyyah, who did not marry. They were occupied from the Sunnah of marriage by what was more important, such as seeking knowledge and engaging in jihad.

And Allah knows best.