True and good point buttttt we don't know that Zee knew Hawkman outside of the League either.
I think thats a problem with the Justice League as a whole. Hawkman does not really know any of them that well. He has not even known them for that long. So his death does not have as much weight for Clark and the others as they want us to feel.
Most of the members of this "team" barely have interacted with each other on screen. It feels more like a loosely affiliated conglomerate or franchise of heroes than an actual team.
I would like to see Dr Hamilton betray the heroes by showing that he is working for the VRA and has bought into their propaganda. It would fit with his other incarnations. In the comics Hamilton became the Super-villain Ruin. Also in JLU he it was revealed he was working for Cadmus.
I think it would really develop that character and the theme of their secrets and with who they can really trust with them. They been way too free and easy with their secret identities. This guy is just an employee of Oliver. Being betrayed by a character who has been around for a while would have an impact.