Let's say you put identical missiles on both. Once in range, both fire. The F-22 can then turn around and get back out of range before it's hit. The Fury is essentially motionless, and will be shot down. Repeat ad infinitum.
Why are you assuming that a hypothetical 1-on-1 scenario is the only feasible scenario? If an operator can man half a dozen of fury squadrons for the same cost that an operator is able to afford a single F22, why would you assume that they would put up a single fury to counter a F22?
And let's assume that a wave of furys managed to down a F22. Now what? Will the operator of the F22 spend an additional $180M to churn out a single unit? Because for the same cost, the opponent can churn out 100-fold units. Is this sustainable?
You can't just divide the cost of a Fury into the cost of an F-22 and assume that's even remotely reasonable as a fieldable proportion. The limiting factor isn't the cost of building the planes, it's the logistics of coming up with with enough fuel, pilots, maintenance, and ground facilities to keep them flying. There's nobody who couldn't afford something better that could afford to field maybe 3:1 vs the F-22.
The salesman for Burt Rutan's little gun-armed midget ARES fighter/attack plane liked to tell people to "imagine how the Falklands War would have been different" if the British Harriers had been faced with waves of hundreds of ARES planes.... completely failing to address the fact that Argentina couldn't possibly train enough qualified pilots to sit in those hundreds of cockpits.
Fuel's relatively cheap for a prop plane, one pilot would manage more than one drone aircraft at once, maintenance is dramatically easier and cheaper on a prop plane, etc.