I can't speak for the later Ottoman Empire, when the authority of the Sultan had been largely sidelined by the government, but during at least the 17th century Sultans were restricted from undertaking the Hajj by political instability.

The Sultan rarely left Istanbul without an army (short hunting trips were usually the only instance where this happened) and spending too long away from the capital invited the risk of a coup in the Sultan's absence. Thus it was a surprise occasion when Sultan Osman II announced in 1622 that he intended to cross the Bosporus with only a few soldiers in order to make the pilgrimage to Mecca. It was widely thought amongst the capital's elite that the announcement was merely a cover for a plan to replace Osman's Janissary regiments with men recruited from the peasants and tribes of Anatolia; Osman's opinion of the Janissaries had soured after a failed siege at Khotin in Poland the year before. In an effort to avert the brewing crisis the Sheikh ul-islam Esad Efendi (the highest religious authority in the Empire) offered the interpretation that the Sultan wasn't actually required to perform the pilgrimage. It is unclear whether Osman genuinely intended to go to Mecca, but the important fact is the (entirely Muslim) military of the Empire did not believe him and on the day that he intended to leave Istanbul, an insurrection broke out that ended with Osman's death and the re-institution of the previous Sultan Mustafa.

TLDR: Some religious interpretations in the Ottoman Empire exempted the Sultan from the requirement of the Hajj, but few were able to try and the one time a Sultan intended to travel to Mecca he didn't even make it out of the capital.

Finkel, C. (2005). Osman's Dream: The Story of the Ottoman Empire 1300-1923. London: John Murray.