No, it does not, the item description ignores the effects of normal rain and harsh sunlight, but the primal abilities are not normal "weather" conditions and are treated as "special weather conditions" and is a special battlefield state. So, desolate land is "Special Weather", not "Harsh Sunlight", and Primordial Sea is also "Special Weather" and not "Rain". Since both battlefield states are programmed to be flagged in this way, the Utility Umbrella item doesn't recognize the "Rain" or "harsh Sunlight" flags. This is why fire moves don't get a boost from desolate land, nor water moves get boosted by rain.