Is the quest leader required to be part of the team?
It is legal to create a team that does not have you on it.  It should be very rare though.  Here's why:

The argument that the previous teams succeeded is not logically strong enough to force the leader to leave himself off the team.  He could argue that the previous teams had a sleeper spy (which is not provable either way with the information public at the time) and replace one with himself.

Consider that a person can always make the argument that he trusts himself more than any other single person, and that argument is essentially irrefutable.  

After all, if he is a resistance member, he knows with 100% certainty that he is a resistance member and should put himself on the team. Therefore, a resistance leader would need a very good explanation for why he would be excluding himself from the team he builds.  

Since a spy should frequently act like a resistance member to be convincing, a spy leader also has a strong reason to put himself on the team and needs a good explanation to exclude himself.

In my opinion the third leader in your game played poorly in that situation.
That player should have replaced one of the earlier team members, had some discussion about it, and put it for a vote.