Logically, with Ra's being more global threat, it would make sense he would offer it to more heroes other than Batman. This is not exactly the same thing. Also, remember Batman does not exist in this continuity or is still an urban legend and not well known. Who would Ra's offer the leadership to? They will probably explain his reasoning in a future episode.
I am not so sure Oliver is going to say no. The show runners have talked about how the ancient medieval definition of "assassin" was very different than what it is today i.e. it did not always imply murder. This Oliver Queen is much more willing to kill. Even after starting his no kill rule, he has seriously considered killing people on multiple occasions. I think he might actually say yes because we do not know how the League operates in this universe. From the surface, it seems like the League is a group of vigilantes with medieval values. Maybe if Oliver took over he could make them more of a force for good. If you look at the trailer, he seriously considers the offer rather than immediately saying no. This plays into his identity crisis and Team Arrow not needing. If Oliver feels no one needs the Arrow or Oliver Queen, he might just decide to do it. You might say they would never do something like that. However, Angle did take control of Wolfram and Hart and one point. Next season is going to be HIVE. What if Diggle and Oliver used the League of Assassins as a weapon against HIVE?
I have read Mike Grell's run and he kills people. I am pretty sure this is were the show runners go the idea of having him kill people in the first season.
Stephen Amell has said he would love to play a more light hearted character and that he does not really enjoy the mangst. During the season 3 première, Oliver is becoming more comfortable and cracking jokes. It Stephen Amell has said that he thinks they are progressing towards that character. I am not so sure about the political outlook part because they probably want to keep politically neutral for a wider audience to enjoy.