The most accurate analysis of why the Ottomans chose the term "sultan" for their leader would be:
The name gave the impression Ottoman rulers were liked by caliphs.
The term "sultan" historically implies authority and power and has been used by various Muslim rulers. By adopting this title, the Ottomans aimed to legitimize their rule and project an image of authority, aligning themselves with the broader Islamic tradition. This title also suggested a degree of recognition or acceptance by the caliphs, reinforcing their political and religious legitimacy in the eyes of their subjects and other Islamic states.