People have a variety of reasons to do what they do. What your grandmother does may be the only way she learned to interact with the world. It is not excusable but it may be the only way she knows.
When you lack physical strength compared to the people you have to deal with sometimes you get used to getting what you want in indirect ways, like lies, plotting, etc. That is why some older women that have lived in very toxic environments when they were young are so toxic themselves. You can't hope to beat down your husband with your fists, but maybe you can get what you need or want in other ways. It is sad, but it is true.
I tell you this so you maybe can find it in your heart not to be so angry with the lady. My grandma on my dad's side was quite the case, she hated me since I was 4 I think, my whole life. She once threatened to hit me with her very solid cane, luckily my mom was there. Every opportunity she had to make me feel bad, she took. She was really mean to my mom too. And I always defended my mom, so that did not help.
In the end, these people drive everyone away so they wind up alone. I learned throughout my life what a horrible life my grandma had. It made her very twisted. Note that I am not justifying her, I am just understanding her.
In the late years, after she managed to do a lot of damage to everyone in my family, she was all alone. Even the people that she had favored left her alone.
My mom took care of her a lot and when she was so sick she was in the hospital my mom went to see her and take care of her there too. She did it for my father's sake. But also because she took pity on this woman that had such horrible experiences when she was young and in the end she was so alone, after giving so much to their children. I am certain that if my grandmother hadn't being alive when I was born, my life would have been much better, yet I did go to see her to the hospital.
I felt nothing but pity for my grand mother, that is why I went to see her. It is funny but in her way she apologized with me and my mom, she said that the people she never would have thought would be there for her, was there for her in the end. It was sad, she could have had such a better life if she would have been nicer to us, instead of this constant little wars and acts of aggression.
Anyhow, you said you base your actions in logic, not emotion. But for me you are not being logical, you are taking the path of most resistance. Going there and saying "there there, get better soon" would be the most logic. I went to see my grandma to help my mom out and to show my dying grandma some support, but I was emotionally unaffected. I did it as an act of charity for a person in need, not for a grandma. Also, my dad did care about his mom, so this also made him feel good.
For me, my core, myself is being polite and trying to be nice. So even though I didn't care for my grandma, as a grandma, she was a human being in pain, so I could relate and try to be supportive. I did not betray myself by going there. By that is how I am. You can be someone that chooses to say "I don't go there because that person is horrible and doesn't deserve me going there". And is a valid position too. Just make sure that you are truly acting that way to be true to yourself and not out of anger or resentment towards her so you don't regret it later.
I never had to say quite exactly what you want to say, I have however told people my feelings about the several relatives that acted this way and I have managed so in a way that well, at least there were no tears, maybe they didn't agree with me, but they understood my position.
As everyone has pointed out if you say something like you want to say you will probably hurt your mom. However, families have shared history. Your mom knows about your grandma and knows about the things she does and knows about the things she did to you. So you just need to tell her:
I didn't go because I fee uncomfortable around her since she has done
X and Y. I don't feel that going there will make her feel any better
and I won't feel better either, because I will remember what she has done and
I will be worried about what she will do in the future by me being there.
This transmits how you feel and why and you actually are not saying anything horrible your mom will be shocked or surprised by. Don't over do it though, the point here is not to talk ill about your grandmother is talking about how you feel. If she insists you can mention something like:
In the past she has said those things that made you feel really bad
and worry about me. If I go there I think she might repeat that. I
don't think it is what you need.
If she continues insisting you just need to spin this same concept around and saying it makes you uncomfortable and will be for the worse. However, I advise you to say something like the following to your mom, and go if she asks you to. But this is optional.
Mom, I don't really want to go, as I said, she makes me uncomfortable,
but if you need me there I will be there for you. I just want you to
be aware she may start talking ill of me and I don't want it to be a
problem for you.
In the end you care about your mom and you don't want her to feel bad. So If you do it, do it for her. You say your mom loves everyone, but she may know things about your grandmother that you don't know, sometimes families hide dark stuff. And maybe that is why she is there. Also as your mom is your mom, her mom is her mom. And that is a bond incredibly hard to break if it was kept though your whole life, no matter how nasty the mother in question grew out to be.
Finally, you don't have to go every day you can go once in a while. Think of it as an investment. A little of your time to make your mom feel better.