There is not only loyalty in that situation. There is also lying to your partner and the children, and the example and values that you give to your children.

Apparently the children are important to you, otherwise you would be gone from a bad relationship. It involves lying to people that are important to you, which does not speak well for trustworthiness. I can imagine the children are more important than someone outside the family, so at least I need to set some expectations right. That doesn't mean trust is a black and white thing.

What would be the alternative to lying?

well the alternative to lying would be that both partners know they can't stand each other but are remaining together for the sake of the children, I figure once you are at that point of remaining together for the sake of the children and nothing else probably both parties know the score.

What if your (ex) partner takes the children away with them? You can't be sure that they also consider staying together for the children to be the best course of action.

Obviosly you are trying to take something that is not yours.

Getting a five finger discount to get (unilaterally) open relationship at a cost of regular one.

What do you mean? I don't understand anything of what you are saying? What am I trying to take that isn't mine? (And why me, I am not the OP)?

What is a five finger discount?

Five finger discount is when you steal.

The condition of staying in good relations with your wife where she does not take the children and blast off is not having a romantic life on the side.

You understand this condition but try to get around it by lying. Not because it is unfair or impossible. Just because you want to have good things without paying for them.

So in your opinion, the punishment for having an affair should be to have one's children taken away from you? Or at least, you think it is an appropriate punishment. I don't think everybody would agree.

You thought all of that by yourself.

There are a lot of ways to breach a "contract" that is marriage, including a voluntary termination. However, being unfaithful is a good way to ruin your marriage, so if it means things to you please don't cheat your wife, unless explicitly being on different terms (and even then...).

This one is totally reasonable. It does not grant you any protection for the other cases, though.