thermodynamics - Is Randall Munroe's "what-if" xkcd correct that magnified moonlight can't get things hotter than the moon?

Note: I am NOT asking if moonlight can be used to start a fire. I am asking whether these particular arguments in support of the claim that it cannot are correct. i.e. I'm looking for an answer that addresses the physics of these arguments specifically.

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On this page, the following claims are made:

1. 
   

   You can't start a fire with moonlight no matter how big your magnifying glass is.

   
   


2. 
   

   General rule of thumb: You can't use lenses and mirrors to make something hotter than the surface of the light source itself. In other words, you can't use sunlight to make something hotter than the surface of the Sun.

   
   


3. 
   
   

   Lenses and mirrors work for free; they don't take any energy to operate. If you could use lenses and mirrors to make heat flow from the Sun to a spot on the ground that's hotter than the Sun, you'd be making heat flow from a colder place to a hotter place without expending energy. The second law of thermodynamics says you can't do that. If you could, you could make a perpetual motion machine.

   
   


4. 
   

   (more claims which I'll omit here)

   
   

I neither believe the claim nor follow any of the reasoning.

First, I don't get the thing about the perpetual motion machine:

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  If this is regarding the first law of thermodynamics (conservation of energy), then it's perfectly possible to lose energy while still heating up an object, which would avoid perpetual motion, so I don't get the argument.

  
  


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  If this is regarding the second law of thermodynamics (increasing of entropy), then it's also invalid because the disorder in the system is still increasing.

  
  


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  If this is about something else, then I don't know what that is.

  
  

Second, here is a video of a guy using a mirror he can hold in his hand to light paper on fire.

Clearly the mirror itself isn't getting as hot as he's making the newspaper, and clearly the mirror is the one reflecting the sunlight.
So how can one claim that the paper fundamentally can't get hotter than the reflecting surface? Why can't the moon's surface fundamentally behave similarly (albeit with poorer reflectivity)?

Can someone explain? Is Randall confusing heat with temperature? Or maybe conduction with radiation? Or am I missing something subtle (or perhaps not so subtle)?