When you are performing with cards, you are demonstrating skill.
It can be portrayed that way, but that isn't necessarily the only way to portrait it. There are card routines that look like skill and those that look like magic.
Skill can be part of magic but magic isn't just a skill.
True. See the first part of the Robert-Houdin quote about being a magician (which you don't have in your post)
Most people, when they ask you to perform for them, they aren't doing it to be rude or think you are trained monkey.
Agreed. I see your point that a true magician should be that 24/7... which means learning magic with everyday objects or being prepared by carrying things to perform with. But my point was that you don't ask a doctor to perform surgery or an accountant to balance your check book -(it does have to be yours because it wouldn't be real if the checkbook wasn't borrowed) if you meet them at a cocktail party, why would you ask a magician to do magic?
When a mentalist asks you to write something down, you can give reasons why but what you are performing will not feel "pure".
That is the view you have decided to impose on yourself. I have no problem crafting a routine where it seems perfectly rational for the person to write down what they are thinking. I've never had any spectator question why they have to write something down. It may be different if you are performing for a single person, but if you are performing for multiple people, it is the only way that the other people can verify that you accurately named what they were thinking of. As a result, if I could really read minds, I would have them write it down as a way for the audience to see that I'm correct.
Also, there are many routines where the spectator is not required to write something down. Thus, it can be justified in one effect and there be no need for it in another. Also, there is a lot of symbolism in writing something down - especially if it is a name of someone, a fear or problem, a question, etc.
You can ask them to choose to think something "random" but from a book isn't logical and doesn't make sense.
It is up to the magician or mentalist to explain (in a plausible manner) why they are using the book. It is also up to the magician to sell the effect.
I was watching a friend's magic and mentalism show. I had helped him develop and script parts of the show. My friend had introduced me to the person who was in charge of the charity that was hosting the show as his "theatrical and technical consultant." I was talking to person in charge the charity after the second show of the evening and she expressed a little bit of sadness that she couldn't experience what the people who were called up on stage experience. I smiled and asked her if she wanted to try something. She agreed. I grabbed one of the three books that had been used in the performance (they were the performer's books that he actually bought at a bookstore when he arrived in NYC). I had her flip to any page she wanted and pick out a prominent word or phrase. I asked her to focus on the word or phrase and to visualize it. I looked in her eyes and then closed mine. I told her "I have a sense that she was thinking about noticing something but that I was also getting an image of writing something down but not actually writing it down but thinking about it." I looked at her with a look that said "am I on the right track?" and she very subtlety nodded. "There is something I'm missing." She bit her lip as if she wanted to tell me what I'm missing. "It is taking something." Pause. "It is taking notes or a note." Pause. "But you are thinking it. It's a.." Pause as if I'm looking for the exact phrase. "Mental note. Is the phrase you are thinking of "taking a mental note?" The look on her face was a combination of amazement and belief. Based on her reaction, she has an amazing story that I suspect she has retold many times. I can almost guarantee, in none of those stories she said, "Yeah, but it was strange that he used this book that was sitting on the nearby table."