Hi, Joe, I know you're not a big fan of guidebooks, but the forum guide, done by Johnnie Walker, is really more of a compilation of the experiences of many of us. I think you'll find it helpful.
https://www.caminodesantiago.me/community/resources/categories/camino-de-madrid.51/
Kinky has a nice report, which I've made a "sticky"
https://www.caminodesantiago.me/community/threads/finally-my-camino-de-madrid-june-2014.25354/.
My Camino de Madrid was a couple of years earlier, but I don't know that a whole lot has changed.
https://www.caminodesantiago.me/community/threads/my-leisurely-stages-on-the-camino-de-madrid.11148/
For me the absolute best thing about the Camino de Madrid is that there is so little asphalt. Infrastructure is great, and though the only real "destination city" on the route is Segovia, that's worth the price of admission all by itself, IMO. I also took a detour to Valladolid to visit the sculpture museum in a beautiful plateresque monastery. There are a few pretty amazing castles, Coca, Manzanares are the two that I think are definitely worth a visit. Medina de Rioseco and Vilallon (two of the end stages before Sahagun) are quite nice towns for a few hours walking around. I think you'll love this camino.
I know you like to hit the ground running, so if you arrive in the am in Barajas, you could take a Cercanias to Chamartin train station and from there you are about a half hour's walk from the first arrow at the Plaza de Castilla. Tres Cantos is about 18-20 flat and easy km and there is an albergue there.
Or, if you are going to go to the Church of Santiago for credential/stamp/blessing, it's in Old Madrid, so Cercanias to Sol and then it's about an 8 km walk from the church to the arrow at Plaza de Castilla, through downtown Madrid. I did that on my arrival day, then slept in Madrid and started walking from Plaza Castilla the next day to Colmenar el Viejo.
Happy to answer any specific questions, there are quite a few Camino de Madrid alumni who have been there in the relatively recent past. Buen camino, Laurie