Challenges of Team Leader in Agile Team

Can you clarify the sort of problems in team leadership you compensate for, and what you mean by assigned versus unassigned work?

 We have best performance in the unit. Scrum Master requires only to have unassigned tasks, while it is important to say that each team member leads his own big project, and I don't really see how it is possible to do all unassigned.

You are looking at this from the point of view of a manager, where you feel you have to manage people  and make sure that they have work.  Scrum approaches it as a group of individuals with the skills needed to produce value deciding how it is best for them to organize to get the work done.  You are describing a situation where each team member is working in a silo on different projects.  There is no team work, just a team working.  Team work is when people collaborate to achieve a goal.  To use a sports analogy, a soccer team works together to put the ball in their goal as many times a possible while also preventing the other team from putting the ball in the opposite goal.  Each member has some predominate skills but they all work together.  Now, take a gymnastics team.  They may work together at times and they hope to, as a group, get more points than other teams. But the entire premise is built upon individual performances.  Each individual is competing against all other individuals, including those on their team.  Scrum is the soccer team, you are encouraging the gymnastics team.

Scrum Teams will take ownership of individual items based upon their skills and availability. But the distinction is that the individual pulls the work from a list of items.  They do not have someone push the work upon them. By not assigning work, the team is allowed to chose their own work and self-manage the work.  They all win together. 

We have best performance in the unit. 

How is this measured? Is it because everyone on the team always has a list of things to work on? Is it because the team members can show how each individual is completing more tasks than individuals on other teams?  There is a difference between "best performance" and "most value delivered".  The value is judged by the stakeholders.  And ultimately, they are the ones that will keep the company in business.  I have known a lot of developers that complete more tasks than anyone else.  But they provided less benefit to the product than other developers that completed half of the amount.