I think your talking about different things.
I honestly don’t know how the pensions were handled. Alberta Martin who was featured in Confederates in the Attic died in 2004. She starting getting a widows’ pension from the State of Alabama in the 90s, when the state realized there was still a widow out there.
As of 2013, two children of civil war vets were still receiving benefits from the VA.
I have no idea how graves for veterans were handled.