I think we are going to go around in circles all day about this. You say the BGA can break out by putting vias all around under the chip and I'm saying these vias create a road block, which you aren't hearing or seeing. The QFP doesn't do that. There is room under the package to route other signals than the FPGA signals, so it doesn't use the entire space you are counting as "used".
There is no road block just from vias as they leave plenty of space between them for 1 mm pitch (enough to route two 0.1 mm traces), unless you have enough signals to fill entire area, but in that case QFP would take more than triple the area of equivalent BGA (BGA-256 is 17x17 vs QFP-208 is 30.6x30.6, or even more if you count entire footprint, and not just the chip itself).
The board I'm shipping now uses a 100QFP which fits on the board with a few mm to spare. Signals need to route under it to get past. By using vias under the package, layers can be changed and still have room to route the signals that need to get past. With the BGA, it might be slightly smaller (a 256BGA just fits, the 196BGA is a bit smaller), but it requires vias densely packed under the part creating a traffic jam that other signals have to route around. Sure, you can use more layers, but that's the point, you have to use other layers and increase the cost of the PWB.
With the 100 QFP there's enough room that I put three 8 element resistor packs on the other side of the board from the FPGA as well as the decoupling caps, about 8 I think.
LOL - QFP 100 is going to have about half the amount of IO compared to BGA-196, and yet still take more area than that BGA. So if you would only route out about half of IO balls of BGA-196, you will have a lot of space under the chip to place all kinds of stuff. AND still end up with using less PCB space.
I've never seen a BGA used without significant space left open around it and nothing on the side opposite other than perhaps decoupling caps.
It kind of seems to me that you haven't actually ever routed BGAs. They are very easy for routing compared to any other package with similar amount of pins/balls, again, taking into account the amount of area those other packages would take.
Oh, and will haven't touched an elephant in the room - namely, horrible SI of QFP packages. That's the reason there are no DDR2 and above memory chips in non-BGA packages. So for anything more complex that LED blinkers you will have to use BGAs, which kind of makes the whole discussion moot.