What to do with it is to act understanding and empathetic with people like that instead of standoffish and hostile. You still insist on the better way of doing things, but there’s no actual need to attack anyone that doesn’t support the better way of doing things, even if their reasons aren’t rational or even morally questionable/bad. It only serves to further entrench them in their positions, while the opposite might have a chance to happen in a more cooperative approach.
It’s relatively reasonable to expect a person to lie, it’s a bit less reasonable to expect two people to lie, it’s even less reasonable to expect someone to lie in a professional context where their livelihood depends on them not being discovered to be lying.
It makes a certain sense when you look at it that way from an employer’s perspective.
Of course, like you probably understand, it doesn’t make any sense after all, because in the end if you go to a doctor and lie about being sick (symptoms), the doctor is neither lying nor professionally liable and the whole thing is just an additional hurdle to go through.
But that hurdle is also part of the point to reduce the convenience of lying. And I’m absolutely sure that this additional hurdle has prevented someone somewhere from calling in sick while they aren’t.
Again of course, that likely hasn’t resulted in more work being done, because obviously the employee had a reason to lie about being sick. But whatever, I’ll stop now.