but I have a strong feeling that something vital is missing, making the picture incomplete, something, that makes a true leader stand out from the rest. As I turn to almost any sphere for examples of a leader, I find prooftoof my doubts.
You should state in the intro paragraph what it is that you strongly feel makes a leader not just good/satisfactory but truly exceptional. You leave the reader hanging by ending it with
As I turn to almost any sphere for examples of a leader, I find proof to my doubts. State what your essay is about (the importance of leadership and public service) to YOU. Don't wait until the end of the second paragraph (where you only mention public service) or the beginning of your final para (the first place where you mention leadership.) Also, writing in positives creates a better impression than negatives.
It doesn't really matter what career you choose, it's more about how you see yourself as an integral part of society.
The first part of the sentence does not connect to the second. Do you mean career is not important if you are an integral part of society. What do you mean by this (integral to what society, in what way?)
For me this feeling of belonging results in a desire to play a certain role in the life of my community and have an impact on the lives of people around me. I'm going to fulfill this desire through public service.
It would help frame your essay if you introduced your concept of leadership - that public service is essential to good leadership - and that YOU aspire to achieving that by giving some good specific examples of your community service.
The ending sentence also leaves me hanging.
To tell you the truth, this relatively simple question gets me confused. The only way I can explain why I'm doing it is because I simply enjoy it, and it helps me feel part of my society.
Why does the question confuse you? Or is it that you have a simple answer to why you love volunteering - you simply enjoy contributing to society. This is more positive (than the word "confuse") and gives you an opening to include the leadership aspect you started out with, thereby making the essay more coherent.