I actually started the Camino Salvador, a historic 7 day Camino that terminates in Oviedo and intersects with the Primitivo which I then hiked in mid-September to Santiago Compostela , arriving there around October 15th. On the Primitivo the weather was mostly dry, the nights were chilly but not too cold except at the high elevations where blankets were available in the albergues. There were few “completos”,meaning no available beds, in the mountains there are fewer towns therefore greater distances and fewer accommodations but the beauty of the landscape surpasses the inconveniences encountered. I do not recall much rain. And encountered very few other hikers until the route intersected the Frances a couple of days outside of SC when it became Like walking through a Central Park - many people.
All in all, the Primitivo in the fall is a must.