What I heard about drafting Joe Mixon - I think it came from Daniel Jeremiah, who is a former NFL scout - is that no team will want to have Joe Mixon as its first pick because that's the face of your draft. I believe he used the Browns as a good example of a team that could get Mixon at #33 and he would be the third player drafted by them.
My current opinion is that it really does not matter where you take Mixon. The team will have to deal with the issue no matter what. I don't think any team can say "hey, we are against that kind of violence, so we only took him on the third day". You drafted him, you will put him on the field and you will celebrate when he scores. If the team has the green light from ownership to take Mixon, why not take him where he would be ranked purely by his talent?
No one in the media is going to ask less questions and leave your team alone just because you drafted him in the 4th round instead of the 1st round.