Is Mama a wholly unsympathetic person?

No, she's actually quite sympathetic in regard to her daughter, Maggie; at least, by the end of the story, she seems to have developed greater sympathy for Maggie than she's ever had before.  Although Mama characterizes Dee fairly harshly -- as a child, "she burned [Mama and Maggie] with a lot of knowledge" and had a "scalding humor" -- her descriptions of Maggie are more feeling, even in the beginning.  She asks,

Have you ever seen a lame animal, perhaps a dog run over by some careless person rich enough to own a car, sidle up to someone who is ignorant enough to be kind to him? That is the way my Maggie walks. 

In calling the person who would hit a dog with their car "careless" and the dog itself as "kind," Mama clearly establishes sympathy for the lame dog, here, a metaphor for Maggie. 

However, near the end of the story when Wangero (Dee) insists that she needs the old quilts to hang on her walls at home -- quilts she had rejected years ago as "old fashioned" -- Mama says that she'd promised them to Maggie for when she got married.  Wangero grows angry and says that Maggie would be backward enough to actually use them, and Mama says that she hopes Maggie does use them, and she can make more when they fall apart.  Finally, Maggie simply offers to allow Wangero to keep them.  Mama says,

[Maggie] looked at her sister with something like fear but she wasn't mad at her. This was Maggie's portion. This was the way she knew God to work.  When I looked at her like that something hit me in the top of my head and ran down to the soles of my feet. Just like when I'm in church and the spirit of God touches me and I get happy and shout. I did something I never done before: hugged Maggie to me, then dragged her on into the room, snatched the quilts out of Miss Wangero's hands and dumped them into Maggie's lap. Maggie just sat there on my bed with her mouth open.

When Mama sees poor, wounded Maggie with her scarred hands hidden in her skirt and realizes how humble and selfless Maggie is, so much so that she can't even be angry at her sister for her selfishness, Mama wants Maggie to come first for once.  She seems to have an epiphany about the value and worth of her daughter, and she will not now allow her to be walked on again.  This description, especially, strikes me as very sympathetic toward Maggie; Mama suddenly understands the emotional toll that being Wangero's sister has taken on her (in addition to the physical toll).