Hi OP, Hope you are still around. I had a similar space, 30' by 25' open kitchen and family room with a 12' ceiling. I bought a used Salk HT3 right / left / center and 15" Sub for about $6k used and run it with a couple of Crown Class D amps with tons of power (almost 700 watts at 4 ohms) and it fills the room nicely and would be at the low end of your budget. Salk makes fantastic looking and fantastic sounding speakers. If you can find some on the used market they are hard to beat. Of course for your budget, if you go the Class D route, you can probably afford to have Jim build the speakers for you and pick out your own veneers. And no, I don't have any relationship with Jim Salk other than enjoying the way my speakers look and sound.
Good luck with your journey.
Good luck with your journey.