I have been the recipient of poor quality/poor quality fabric stuff. I currently have a pile downstairs of baby tops I'm not sure what to do with. I do know that I won't back them and give them away--they are really, really ugly. And I can work with just normal ugly.
So, what I'm thinking I will do, is cut them into smaller pieces and use them for a QAYG larger top and use a snazzy sashing and repurpose the tops this way. I will make several of these small ones into one larger top. I think they will be acceptable that way.
Otherwise, I've finished double knit tops given to me, and they work fine.
I've also had some "squares" given to me that weren't even close to square and the seams completely uneven (so I understand more or less where the OP is coming from). It is sometimes a challenge to use what is given and make something pretty/nice out of less-than-ideal materials. In my case, I don't have a clue as to who is doing the sewing. It would be hard to be gracious if it was a constant thing (IOW, the seamstresses think they are doing something wonderful when they are giving you garbage.).
Definitely, try to talk about quality and what makes a good charity quilt. Some people are still of the understanding that "missionaries" still want used tea bags, too (I guess.). I suppose they think that for charity, "any old thing will do." I try to make everything I give away (with the exception of the double knit tops that were donated to me and I finished) something I would be willing to have in my own home.