Should we prune our large ash tree?

no. mulch to the drip line and reroute your sidewalk if possible

Pruning the top of the tree will make no change to the growth of roots. Surface rooting is caused by the moisture available to the tree being available easily or only on the top levels of the soil. Deep watering will encourage the tree to seek moisture at a lower level. This is not a sign of disease and cutting down a mature tree can cut thousands from the value of your property (curb appeal).

No tree is perfect but the value in trees is in shade, colour, & ambiance (this is aside from the air purifying benefits). If you have to replace a sidewalk this is surely a small price to pay for being able to own and enjoy such a marvellous amenity.

Your best bet is to consult an Arborist. They have the expertise to tell you what to do with the tree. It is possible that there is some disease process going on. Hopefully it is not Emerald Ash Borer.

They would also be the ones to remove the tree if need be. Just make sure the Arborist you hire is insured. That way if the tree falls on a house while being removed, THEIR insurance pays for it, not yours.

Pruning the tree will cause the roots to grow even more. If it were me, I would remove the tree, as it is only going to get worse.

it relatively is recommended to hunt for suggestion from a community arborist in this one. Emerald Ash Borer is a controversy interior the midwest US, yet i'm not acquainted with them being interior the southwest. Have them examine your wood and in case you have EAB, they are in a position to handle the wood to get rid of the bugs. If that's something else, they could be able to deal with it besides. stable success!