I would be careful about asserting yourself more with your DIL. It's only likely to push her further away if there is already tension of some kind.
As far as once a month being 'normal', what is normal will vary from person to person but once a week is, IMO, exhausting. I do think that you are wanting too much and are definitely not obstructed from seeing your GC at the rate of once a week. Whatever you did with your children was up to you, it doesn't mean your children will do the same for you.
I would be careful of things like this. You wrote: He tells me that she’s upset because I said this or that but I can see she’s twisting my word
So you know she is upset because of something you have said but you are trying to dismiss it. Your not accepting it doesn't change the reality of what you have been told and how she feels. Even if she is being sensitive, and I don't know if she is or not, it's how she feels.
Also, you shouldn't get involved in their arguments. It is interfering. No-one should be 'in the middle'. If your son is in the middle, he needs to stop trying to keep both parties happy and remember that he forsook all others when he married his wife. That doesn't mean you aren't important, it just means his wife is his primary relationship now. Example: If you want to take the baby overnight, DIL doesn't want that, your son should not be in the middle trying to calm both sides. He should be supporting his wife and telling you no. As hard as it can be, when our children marry, we are no longer their primary relationship and the wants/needs of the mother trump the wants of the GM.
I think you need to look at your own part here a bit. That's not to say they are perfect, of course they aren't, but if you re-read your post, try to see how you are pinning all the responsibility for the perceived problems on your DIL and dismissing what they have told you.
There are a lot of clues in your posts. Even your first post. You say you have to 'push and push' to see your GC. How do you think your S and DIL feel about constantly getting pushed and pushed for access to their child? Your DIL is not answering the phone when you call to see if you can see your GC. What does that tell you? I suspect she isn't answering because she doesn't want you constantly visiting or pushing them.
None of this means your DIL doesn't like you. I don't think she is coming across unfavourably in what you write. I do think she is coming across as someone who needs some space from such frequent visits and being pushed for them.
In the interests of a better relationship, can you discuss with your son what a suitable visiting schedule would be for them? Make it about their needs rather than what your ideal is.