Every time I hear "dogfighting is obsolete" or "the F-35 will kill its opponents at range with missiles", I wonder. What happens when two stealthed 5th Gen fighters with good ECM try to kill each other with missiles?
One kills the other with a missile. Countermeasures turn the marginals into misses, they don't turn off a solid kill shot; likewise stealth is better thought of as reducing the effective range and SNR of enemy radar, not making you invisible.
Dropping the engagement range from 100nm to 50 doesn't make a dogfight. The strength of radar returns go down with the 4th power of distance so it takes a lot of work to decrease detection ranges substantially.
Forgive my unfamiliarity with the topic, but are these planes really using radar during a battle? I would have thought that anti-radar missiles would be standard equipment?
This is where things start to get complicated but yes, planes use radar. In many cases, though, it's provided by an AWACS that sits back out of missile range.
OK, in that case I see the point of fighters as a secondary line of defense for the AWACS. If the AWACS can stand off so far, however, then the primary weapon could be just a mobile missile platform, relying on the AWACS for targeting info. That seems easier and cheaper to build than whatever the F-35 has become.
To a certain extent no-one knows for sure, because there's never been an all-out war between modern air forces. The go-to example for a lot of the doctrine is the Falklands war which was over 30 years ago. That said:
To a certain extent a modern fighter plane is just a missile platform. You want your missile platform to be fast and stealthed, and you want it to be able to fly close to the ground to make it harder to detect, which means making it maneuverable and at current technology levels probably forces it to be piloted. You want the survivability stuff - ECM, flares and all that.
At that point you're most of the way to having a modern dedicated fighter (like say the F-22). Could you leave off having detection equipment on the fighter itself? Maybe, but that would severely limit where you could operate - AWACS planes can't really operate in disputed territory (they tend to be modified civilian airframes, they don't have any of the things in the previous paragraph, because they want to carry large bulky radars). Radar and the like is a relatively small addition and opens up a lot of other mission possibilities - either escorting bombers in hostile territory or, these days, taking on a bombing role itself (modern fighters are large enough, and modern missiles heavy enough, and modern precision bombs light enough, that it seems like every modern fighter has at least some ground-attack capability). Interestingly enough the F-22 goes in the opposite direction - it's been described as a "mini-AWACS" and can provide targeting information to friendly aircraft.
Is it then worth using the same plane for close air support? It's arguable but IMO that part also makes sense. It's the STOVL requirement that really kills the F-35 - cut the lift fan, narrow the fuselage and you'd have a good aeroplane (which would still have software problems, but I'm not sure building multiple aircraft would have gone any better on that front).
Anti-radiation missiles exist but Air-to-Air missiles aren't widely used. A fighter's radar system is designed to cover a "narrow" area in front of the aircraft. If the fighter turns or if the pilot narrows the radar's sweep, the missile loses it's homing signal. Ground based systems and air based systems like AWACS scan 360 degrees so they can be targeted from any direction.
