Thank you for explaining, this makes sense.
It doesn't matter whether sponsor is working or retired. It matters that your brothers are not citizens yet nor they earned 40 quarters of SSA. Because of this, any living sponsor, whether retired, employed or unemployed are required to file I-865 when moving. I-865 is about sponsor's address change. If sponsor moves, they must file this form.
If your mother still lives at old address, she indeed doesn't need to file it.
Good luck!