David saw one season of the show and got the idea that strong guys never win, and that everyone else should just hand him the win so he could make history. Eva doesn't work well with women and is used to being on a team of boys. She sees things black and white and has trouble detecting lies. The "honor and integrity" alliance just made the game easier to navigate with her autism. Joe had a bit of a holier than thou attitude, and he tried to spin a narrative to justify every move morally.
There's also no inherent connection between strong guys and "honor and integrity". This mindset was illogical and doomed from the start. They either had to break their own rules or it would cost them the game, and we ended up seeing both. I see no reason not to just embrace the cutthroat game you signed up to play.