The reality is that if you arrive in Madrid on the evening of August 31, you probably won’t start walking until September 2. There are trains and buses from Madrid on a Saturday evening, but as Trecile says, it’s a 5 hour journey, plus transition time, so if you’re coming from Philadelphia, you won’t have slept for 2 days if you ty to start walking on Sunday, September 1. You can find bus schedules from Madrid to Oviedo at
https://www.alsa.com/en/web/bus/home. Train schedules at
http://www.renfe.com/EN/viajeros/index.html. Might be better to spend Saturday night in Madrid, then take a bus or train the next morning. Or spend Sunday (when you wake up) exploring the lovely city of Oviedo.
If you start on September 2 from Oviedo, your realistic best case scenario is you arrive in Santiago on September 15. That would mean averaging more than 15 miles a day through very hilly country. Only you can decide whether that’s realistic. You could probably get a flight or train out of Santiago to Madrid late on the 15th, and certainly on the 16th, to then catch your flight back to the States on the 17th, but it wouldn’t leave you much time to “smell the flowers.”
Another possibility would be to just walk from Oviedo to Lugo (also a major city), which is 100 k / 62 miles before Santiago, and from there return to Madrid. That would make your entire journey more leisurely. Lugo is a fantastic city, and makes a fine end point, too.
If grabbing the cheap flight is a priority, then go for it. Worst case scenario is you don’t get to Santiago, but there’s nothing sacred about that.
It’s all good, as my son says.