Working with a multi-generational farm is
extremely challenging, and it sounds like you're making a pretty good go of it. Been there, done that..... The best way to get those chickens into your care--or at least into a better situation--is with a little psychology. Their (relatives') pride has been hurt, and the best way to reach them is to appeal to their pride. Different ideas:
"you have
soooo much else to do, you should let me deal with those stinky birds..." "That coop is getting a little dilapidated, don't you think or it's a bit of an eyesore with the rest of the property or ..? We could take it out and put in...an new one or another garden area or a patio/BBQ area or fire pit (using the old timbers

for family gatherings or fruit/nut trees or workshop (esp for hobbies/therapy). "We could build a bigger better one so you could get
more eggs or, better yet,
I'll build a big one and give everyone some eggs and no one would have to worry about the chickens ever again or 'that would give you more time to.....hobby/visit/.....' Also showing them the result of the neglect (without actually calling it that) may help "it decreases their egg production or ..."
Based on my own experience, I worry that an approach that is too confrontational may create a rift in the family......including yours....