Another cheer for Benduenos albergue if you find yourselves on the Salvador! A detour from The Way and well worth the climb. Great memories - here's some of what I wrote at the time. The descriptions that had lured us there were well warranted. Our stay there was a highlight of the Salvador (in 2018).
Day 4 – Pajares to Benduenos, 15 kms plus an extra 1.5 up a steep road to the albergue. Another tough day of climbing, at times through rough bushland. But a great adventure. Tricky descents on narrow rocky paths made for slow going. We didn’t see anyone else – though they could have been ahead or behind. We chose to add another 1.5 kms to the end of our day (and following morning) to stay at the Benduenos albergue that we’d read about – described as a sanctuary set in a peaceful spot alongside a beautiful church and managed for the past two years by Sandra, with rave reviews of her kindness, hospitality and cooking.
I'd add that if you decide to take the Salvador, be mindful of distances. We typically walk around 25-30 kms per day and that's a comfortable distance. On the Salvador, we walked a couple of 'short' days of around15 kms and it seemed to take almost the same amount of time and effort as our typically longer days.
PS Just noticed you mentioned you'd appreciate private rooms. From what we saw in 2018, private rooms could be a challenge on the Salvador. Others may have more up to date information. It shouldn't be a problem on the Primitivo, depending on your stops. When I was last there (Oct 2019) some of the albergues had both private rooms and dorms - and in some towns there were private hotels/casas. Gronze is a good source of info on options for stages and accommodation.