What does an employer want to see from recommendation letters
I think all 5 points are valid. I would add two more:


Due diligence: Institutions want the insurance in case something goes wrong with the faculty in the future, e.g. the person commits scientific fraud or sexual harassment. Of course, there can be nothing in a recommendation letter that would guarantee that the faculty will behave, but in an inquiry, the institution must be able to say they did their due diligence. So, they just need a certain number (typically three) letters on file, not necessarily for their content.
Intangibles: They want to see enthusiasm in the recommendation letter. I have heard many times that a merely good recommendation letter is a terrible recommendation letter. If someone's adviser/colleague cannot be bothered to write superlatives for the candidate, it might be a "red flag" that there might be a something intangibly wrong with the candidate.