The Ottomans chose the term "sultan" for their leader primarily because it conferred a sense of authority and legitimacy. The term "sultan" originally means "authority" or "power" in Arabic and was used to denote a ruler with significant political and military control. By adopting this title, Ottoman leaders were able to establish themselves as powerful sovereigns in a manner that resonated with the existing Islamic governance frameworks.
While the statement "It was given to the Ottoman leaders by Muhammad" is historically inaccurate (as Muhammad did not assign the title), and "It was the name of the spiritual leader of Islam" might be misleading since the title has different connotations than that of a caliph, the title "sultan" effectively highlighted the rulers' authority without necessarily tying them directly to the caliphate.
Furthermore, while it might sound important when spoken, this alone wouldn’t be a key reason for its adoption; the significance lies more in the political and historical context. Ultimately, the use of the title "sultan" framed the Ottoman rulers as legitimate leaders within the Islamic world, enhancing their status and power among their subjects and rivals. Thus, the correct perspective is that it helped establish the authority and legitimacy of the Ottoman leadership within the Islamic framework.