could use mushrooms. they eat dead wood. Drill holes into it and pack down spores and see what happens. At worst, nothing will happen.
I'd use hugelkulture if it was a good spot for hugelkulture. Or a
compost heap.
Basically you want the stump to return to being a productive element in your system. There are many ways to do that which don't require much work.
I would only take a stump out if I really needed to reclaim that piece of ground. I've done it in a small yard where space was an issue and I had a magnolia that kept sprouting and creating a bush. Lots of digging and cutting, but I got it out. Wouldn't do it again unless really necessary.
If it's just dead and nothing's coming up from it, you could also just choose to grow around it. In time the spot will become something else.
Also, dead wood that is not breaking down needs nitrogen to break down.
Fire is an option, but why waste good carbon?
W