If you arrive in Madrid on the 28th and leave immediately (same day, which means no real free time to see Madrid) for your starting point that means you could start on June 29 after your travels. And if you have to leave Santiago on July 10 you would have to arrive in Santiago on the 9th. That gives you 10 days on the Camino with no free time at either end.
You can start in Ferrol for the English Route, or Tui for the Portugese Route or Sarria for the French route and finish in 5 days (or more if you choose), and still earn a Compostuela. That would give you the option to spend a day or so in Madrid, or Santiago, and still walk the three days to the coast. If walking to the coast is not important, then you could start a further back on the French Route or the Portugese Route. The English route starts in Ferrol.
To find out about the Portugese Route check out this blog - magwood.wordpress.com -. She also did the French Route several years ago. Tui looks like a really nice town to visit.
I did the English route because my time was limited and had a very memorable Camino. The route is supposed to more pastoral than other routes (although it doesn't sound much more pastoral than the Portugese Route). I was alone, by choice, most of the time.
Not having a free day in Santiago to see the city and truly experience the Cathedral would not be an option for me.
The French route from Sarria is supposed to be quite busy, but conversely has lots of options for lodging and food - much more than was on the English route. There wasn't any real lack of accommodations on the English route, just not as many choices, or choices as frequent as on the other routes.
If you do decide to do the English Route, get John Walker's guide book. It is excellent.
Buen Camino