If they really feel strongly about the "...with integrity" part, they should be filtering out some players every now and again with a documented questionable past.
However, if you are a voter and Cam Newton is on the ballot, I don't think you can be the character-judgment police and vote for someone else simply because you think he's a shady character. For all we know, they all are shady characters. Point being, you can't make that judgment because you simply don't have the insight on all the player's lives to build that into your opinion. Voters should vote for what they are best able to judge - athletic accomplishment on the field. They should leave the integrity piece for the folks in the club that put these guys on the ballot at the end of the year. If they want to get tougher, they should do it. If not, they can enjoy their aware becoming further tainted over the years...
That being said, if he were on the ballot, I'd vote for him.