I really don't think bias binding would help with the mitered corners or joining the ends at all. It would be very helpful on curves, but otherwise, straight binding is fine and less stretchy. There are so many binding tutorials out there, and tools to help. Not sure what your exact problem is with it. I'm not sure, either, what the square binding is that your grandmother did to finish her quilts. I think it would be just fine for you to do yours that way if it's easier for you. Just call it your family tradition. LOL
The sewbaby blankie buddies sound like a fun project, and a good way to practice binding. Good luck!