I would bet a large sum of money, this is a simple case of the roots of a large tree are rotting, and that is all.
I do not know how large the tree was but multiple roots over a foot in diameter go down over 4 feet, some well over.
All you did, or whom ever did the tree, was remove a little of the top of the tree trunk probably not touching the roots.
I land-scaped for years and never, ever, saw rats live in old tree roots (now maybe this area has a weird rat problem, so I could be wrong but at that , Rats, do not tunnel under ground.)
The ONLY way you are going to solve this problem is to hire some one with a back-hoe to dig down at least 6 feet and remove ALL the old roots.
They will be decaying for most of the next decade, or longer.
I do not know what rodents you have there but I have had them live in old holes in the ground for decades, some of my dogs and cats would stand by the hole and stare or , in the dogs case go bonkers dancing around the hole sticking his nose in.
They are harmless.
For the decaying wood to be moist and crumbling is just the natural case of rotting wood, depending on species of tree, determines how long it takes to rot. ----( I once buried some old Elm root from a Elm that had been gone for years but I dug out, by hand, the then rotting roots down three feet to reduce collapsing soil, some was not lumber hard but far from crumbly. I buried it in the garden about two feet down. When I planted some rhubarb in that are, years later the Elm wood was mostly still there.)
What you have is the main reason if any one cuts down a large tree and does not get rid of
ALL of the roots at least five feet down, may or will , have pain in the arse years on down the line.
Addendum:
At that, where I put in a garden thirty years ago, twenty some back, I woke up one day to find, a hole over six feet deep and five feet wide.
