I'm not sure if I buy this completely. You see, a manager is human after all which means he/she is going to like some people more than others. There is always some kind of relationship with all the others, no way around it. Now obviously if the manager is only going to look at this relationship, his/her personal affection for others and not their work, and give some advantages over others based on that relationship, yes that is bad. Seeing that, and seeing an actual romatic relationship is just a more involved type of relationship, 'simply don't sleep with the people you manage' is obviously not enough. You should be able to put all or most personal things aside. As such I think these rules of no romantic relationships are a bit strange. Even if you follow that rule, but for the rest still manage people based on your personal relationship with them, it's still bad.
Or looked at it the other way: If two people have feelings for each other but suppress resulting actions due to company policy, how would that not influence decision making?
The first is unavoidable, and although we try to correct for it in internal hiring/promotion processes by referencing assessments from multiple persons and making a data-driven decision, the second is completely avoidable by not dating the person!
After having graduated high school, most adults don't just fall into relationships. It is reasonable to assume intent is present once the connection between two individuals can be considered a "relationship" in this context.
This is why good managers don't aim to be friends with the people they manage. You may end up being friends because of team camaraderie and the amount of time you spend in the same building, but deliberately choosing to socialise with some (but not all) of the team is a bad idea.
And exactly the same thing happens if there is a perceived friendship in the team. Was the employee promoted because of their skills or because they're mates with the boss? Were they not promoted because the boss didn't want to be seen to be promoting a friend? Did they get into that argument because they're a brown-nosing jerk that thinks they're untouchable because they go for beers with the boss every Friday?
Of course not. If a manager runs a tight ship and treats their team fairly, then there shouldn't be anything wrong with a relationship. If as a manager you can't properly separate your emotions and friendships from your management duties, you aren't going to make a good manager anyway. The manager who is going to unfairly promote the person they are seeing is also going to unfairly promote the worker who they are best friends with over the worker who best deserves the promotion.
According to the CEO logic this should not be allowed and would not work. It does. Are teachers better managers than CEO's? Apparently. But then they should get paid accordingly.
Why would you think that bringing a gun to work is a "mistake"? Does one's right to self-defense stop the moment they cross the threshold of their office door?
How can you argue or prove any relationship between boss and subordinate is consensual?
If she got promoted, would it be because of her work, or her relationship with the boss? If she didn't get promoted, would it be because of her attitude, or her relationship with the boss? If she had an argument with a colleague, was it because she was an entitled slut? If he cut the budget of the department she worked in, was it because he was going off her?
It's pernicious and there's nothing you can do about it except simply not sleep with the people you manage.
If a CEO of a blue-chip like Intel doesn't get this basic rule of management, what other basic mistakes is he going to make? Bring a gun into work? ;)