The original question has many implications. If they are not part of our religious beliefs then why should we hold them to it? Righteousness is defined by various religions and individuals. Politicians lie a lot, sometimes it is to protect the truth, is that righteous? Some say yes it is and some say no it isn't so the question is loaded.
I understand what you've just have said, however, I think not all politicians lie a lot though we say most or many of them do. We can't judge them all because we do not know their hearts, only God knows. People in the world for example if there's a new president, most of us criticize them for what they did or judge them without any basis. But think of this, can we find a perfect politician? No, there is none. Even God doesn't require a perfect politician especially if they are not member, but He look at what they can do to us. Hope you got my point. If we keep on criticizing those politicians, and if I am the President, I would rest a while for one year and would say to the people "Ok, you rule the country, let's see what would happen in one year". I ask you, what would happen? Hope you got my point.
My reply is in answer to the question of secular leaders being righteous. My answer is that it is open to debate because being righteous can mean so many different things to people. Will I like my leaders to have temple recommends - sure, should I expect them to - no.