Stump grinding, or leave the stump? An honest comparison
Grinding is not automatically the right answer. It depends entirely on what you want to do with the space.
What grinding actually is
A grinder takes the stump down below ground level using a rotating cutting wheel, turning the wood into chippings. It does not remove the entire root system: it deals with the stump and the roots immediately around it, usually to a depth of somewhere between 15 and 30cm depending on what you need to do afterwards.
Full root removal by excavation is a different and considerably more disruptive job, and it is rarely necessary for a domestic garden.
When you need it done
If you want to lay turf, plant in that spot, build over it, or run a mower across it, the stump has to go. Grinding is also the answer if the stump is a trip hazard or simply an eyesore in the middle of a lawn.
It matters more than people expect if you plan to replant close by, because a large stump left in place competes for space and takes years to break down.
When leaving it is perfectly reasonable
In a wooded corner, a border you are not touching, or anywhere out of the way, a stump is not a problem. Left alone it decays slowly and supports a surprising amount of wildlife. It costs nothing to leave.
The one real argument for removing it
Some species send up suckers from a stump left in place, and poplar, willow, cherry and lilac are among the worst for it. If you fell one of those and leave the stump, you can spend years dealing with shoots appearing across the lawn. That is the case where grinding pays for itself.
Honey fungus is the other consideration. A decaying stump can host it, and where it is already present in a garden, removing dead wood is sensible.
What to expect afterwards
You are left with a hole and a large volume of chippings, and there are always more chippings than the hole will hold. Agree in advance whether they are being taken away or backfilled. Fresh chip is not soil: if you intend to turf or plant, the hole normally needs the chip removing and topsoil bringing in, otherwise the ground sinks as the chip breaks down and the decomposition draws nitrogen out of the soil.
Access matters here too. Grinders vary in size, and a narrow side gate decides which machine can reach the stump, which affects both price and how long it takes.
If you want this looked at properly, Willow & Oak covers Nailsworth, Horsley, Amberley.