Why would a red picker ever lie about it? If I can get all 8 of my fellow players to pick red then we’re all safe. If it’s a button I’ll just break the blue button or wire it to red.

A selfish player will claim that they will coordinate with the group, and then vote red in private. A coordinating player will pick what the group chooses, whether that be red or blue. You are talking about a coordinating player here. Yes, in this case if all players agree to red, it's obvious you should all pick red. It's completely safe.