The ability of the F-35 to "dogfight" is really irrelevant, it simply isn't designed to do it. Of course it failed to keep up with an F-16, a plane that was designed to dogfight a MiG

This. Dogfighting is obsolete. It's possible a fighter will get into a dogfight but it will be an anomaly. The strategy of air power currently is to fly an enormous radar system into the air, AWACS, and then that is the only thing shooting easily visible radar. It's hundreds of miles from the front lines. The fighters go in passive, the AWACS sends them to the enemies and they pop radar on a split second before they fire. There is almost no warning and it's brutally effective.

That is why they want fighters that can't be picked up on radar. If the enemy is lucky they might spot our fighters a little earlier if they're using active radar. (it's like using a flashlight at night) They can sure as hell see our AWACS they just can't hit it and they know there are fighters between them and the AWACS.

The AWACS with a dozen F15's has been completely dominant in air engagements up to now. Add the F22 and their hope is that those fighters will be almost impossible to spot even for air defenses on the ground and the enemy will simply see their planes start exploding in the sky and bombers like the F117 and B2 mean they can also make things start exploding on the ground with very little danger from air defenses.

In order for dogfighting to start mattering again there would have to be some new technology invented that spotted aircraft reliably and neutralized the "stealth" features. Even then missiles would take out most targets long before dogfighting was possible and it'd probably be cheaper all round to just use dozens and dozens of drones. You can send drones on suicide missions as well. Sort of like a cruise missile but it can just sit up there and wait.