What is the purpose?
Traditional Gazpacho doesn't have boiled mashed eggs in it
Salmorejo does, it is sort of Gazpacho with eggs and ham
Salmorejo is creamy and mostly blended, but the chopped or coarsely mashed eggs and their flavouring (garlic and salt), and the chopped ham are added last to give a chunky texture
People seem to use Gazpacho to refer to any cold soup?
Salt acts as a mechanical agent to help you make a paste out of the garlic. It prevents the garlic cloves from slipping against the mortar walls, and helps with the grinding and mashing. Salt+garlic is a common start for a number of mortar/pestle recpies in Spain.