As you have discovered, weed and grass killer does not kill fungus. Zero only works on green plants.

Mushrooms are just the fruiting body of a very large underground network; the network feeds on decaying organic matter and may cover several square meters. Is there a large piece of wood underground in that area? Did you take out a tree and leave the roots? If you want to get rid of the mushrooms you will need to dig down, discover what they are feeding on and remove it. Mushrooms will not attack or destroy a lawn, but if the lawn dies from another cause they may live on the decaying roots.

Since mushrooms do no harm and will disappear once the food source is used up, most people just live with them.