A sword is only as sharp as the one who wields it. The sharingan was nothing but a ninja tool that Kakashi used. And he was good at it considering the fact that he was not a true Uchiha.
However, the tools a ninja uses is not the only criteria that would be considered while appointing him to a post of a kage. His diplomatic skills, his relation with the other villages, his services to the nation are some of the other (among a larger list) that needs to be considered.
In all these aspects, Kakashi has excelled. There can be no doubt on the fact that Kakashi was capable of fighting with and without the sharingan. His skills in strategy were at par with the best of the best. He had fair diplomatic skills, which we can assume from the fact that he had already been appointed for the post of hokage earlier (right after Danzo dies). He was a person of calm demeanor and not pragmatic. He weighed his options before making a decision and could think calmly in the face of adversity. All these does make him a strong contender for the position of Kage.
Tsunade was not only appointed as the hokage because she was a great medical ninja, but also because of her services in the endorsing the idea of the 4 man ninja team with a medic ninja included in each team.
Also, the sharingan that Kakashi had only allowed him to read the movements of his opponents in a precise manner. His ability to copy the jutsu was aided by the sharingan, but in the end he had to commit it to memory so that he could use it later on. Also he must have had excellent chakra control to read the hand signs of a jutsu once and copy it immediately. Also, he himself mentions while aiding naruto create his rasen shuriken, that he could use chakra of elements other than thunder. We have seen him use water style, earth style and fire style. So, his strength is not solely because of the sharingan. Like a tool, the sharingan helped his cause.
