A stump might still be alive. if so, it will resprout new growth eventually. It's not the end of the world if it does, you simply cut back the new growth again. Burying or covering a stump will reduce the likelihood of resprouting and may accelerate decay.
When it is possible, I'm happy to let a stump decompose naturally, but if it becomes a grievance for any reason, it can be removed by any of various mechanical and/or chemical means.
I lean towards the use of mechanical means such as a stump grinder, as I don't like to make a chemical 'hotspot' in my garden with saltpeter (potassium nitrate), epsom salt (magnesium sulphate), potassium hydroxide, acetic acid, etc.
However such chemicals are used often, and presumably the land will recover, or mostly so.