There is some indication that I am not that good at teaching – should I still strive to become a professor? Since I was an undergraduate I have been a tutor many times, either a drop-in tutor or a private tutor. However, somehow these students politely turned away from me for reasons I do not know. In all conscience, I am very meticulous in preparing lecture notes and every class. I thus am tempted to think that if it is possible that I just have no that kind of teaching charisma. 

Is it still possible for me to become a professor?

Please first let alone the other various necessary conditions for becoming a professor.
There are many possible interpretations of why the students turned away from you. You should not jump to the conclusion that you are bad at teaching. For example, it is possible, judging from the information you gave in the comment, that the students simply did not wish to have a tutor, despite the good intention of the parents who hired you. This is especially common in certain East Asian countries where competition for educational resources are fierce. Moreover, depending on the type of students you tutor, your experience may not speak to your potential as a professor at all. After all, the academic need of a high school student is quite different from that of a college and graduate student. Your teaching style may work well for college students, but not so much for the high school ones.

Finally, even if teaching is not your strongest suit, the experience you gain as you finish grad school and post-doc should help you build the necessary skills (yes, teaching can be an acquired skill, not just an innate charisma). So depending on where you are along your education, there may still be time for improvement. Also, depending on the type of university you work in, the relative importance of research and teaching can vary considerably. So even if everything else fails, working in a research-oriented university should help a bit with the problem.