We have a similar climate temperature wise here, but the huge difference is that we get rainfall all year round. Summers do tend to be drier and we can go 4-6 weeks with little or no rain in the hottest part, so that's kind of Mediterranean. Olives are grown commercially in this region and the orchards have a pasture or ley understory. Most growers graze sheep to keep the grass low, although some will go at it with a mower. I've never seen an orchard here that was tilled.
Even though the prevailing tradition in some regions is to till for weed control and (according to some practitioners) reduce moisture loss, it's extremely damaging to the soil life and carbon reserves, and hence harms the fertiliy. The other main effect over time is elimination of structure, which creates compaction, loss of aeration, and reduced water infiltration. So when it does finally rain, there's more runoff and erosion.
I understand that this is a culturally embedded practice, but if you can buck the trend by demonstration, by all means do it. Letting grasses and herbs grow will provide shade and reduce evaporation, even when the plants die off or go dormant for the summer. The root mass and surface litter will hold the soil together when it rains, and help increase the organic matter. These factors will promote all the things you want to see in living soil: more nutrients, more critters, and more tunnels for water to penetrate deeply where it will do some good.
The prunings can be made into biochar to do even more good for the soil.
Example of how olives are grown by a friend of mine
I've been working with Huhana to make biochar from the huge amount of prunings from the trees. Part of the biochar gets spread back on the orchard and we're using the rest to clean up the stream.