What is the "non-apologizing rule"?
I suspect there is no such rule. I looked at a few online sources - none mention the word apology (or variants).

ATC communication is expected to be short and follow a standard set of stock phrase forms.

Anything outside that is discouraged unless necessary to clarify instructions etc.



FAA

An FAA document "Radio Communications Phraseology and Techniques" says


  Brevity is important, and contacts should be kept as brief as possible, 


So any communication that doesn't follow one of the standard patterns is discouraged. It continues.


  but controllers must know what you want to do before they can properly carry out their control duties. And you, the pilot, must know exactly what the controller wants you to do. Since concise phraseology may not always be adequate, use whatever words are necessary to get your message across.


ICAO

An ICAO document "ICAO Standard Phraseology A Quick Reference Guide for Commercial Air Transport Pilots" says


  Phraseology has evolved over time and has been carefully developed to provide 
  maximum clarity and brevity in communications while ensuring that phrases are 
  unambiguous. 


Again the emphasis is on brevity and standard phraseology. It also continues ina similar way to the FAA document.


  However, while standard phraseology is available to cover most 
  routine situations, not every conceivable scenario will be catered for and RTF 
  users should be prepared to use plain language when necessary following the 
  principle of keeping phrases clear and concise. 


Other

For non-pilot rubberneckers like me, a good intro to pilot-ATC communication is a "Say Again" blog post or Two by Don Brown.


  I remember one time when I was a young controller and working a slow, high-altitude sector. I was bored so I got into a conversation with a pilot about something or other. All of a sudden I noticed this F-15 was in a big turn.

 "Peach two one, Atlanta Center, where you goin'?"  
 "Atlanta Center, Peach two one is declaring an emergency, 
        we've lost an engine and are returning to Dobbins."  

  
  He couldn't get on the frequency because I was having a conversation.