Yes but they cannot use the missile in the same way. It is the aircrafts radar that controls this missile, not the attack management radar. The greater the power of the radar the longer the range. That was the whole point behind the Euro fighter and its advantage over everything else out there besides the f22. The F16s radar is quite weak compared with even the f35s, and the newest upgrade is still not a phased radar.
When these shorter range radars use these missile it is not in a BVR capacity. I should also point out that there are counter measures in the works for these missiles that may seriously degrade their effectiveness in a decade or so (laser destruction of flight control, that sort of thing.)
SO we could we be back to dogfighting soon enough. Also, you seem to thick that we are the only ones with large-scale attack management capabilities. The Russians and the Chinese have them. With the new Israeli technologies, India has better AWACS the we have. Besides, in a defensive scenario, one does not necessarily have to have air-based attack management assets - it can be done quite well from ground based radar. In any even, UAVs will certainly make air based attack magament and tracking much cheaper in the future.
The AMRAAM is an active radar missile. The shooting aircraft provides an initial cue for the missile, but the missile guides itself to the target.
"The greater the power of the radar the longer the range."
An aircraft's radar has no impact on the range of its missiles. If that were true, than an AWACS would be the world's most effective missile launch platform.
"The F16s radar is quite weak compared with even the f35s, and the newest upgrade is still not a phased radar."
Yes, the F-16's radar is weaker than an F-35, but the newest version is actually a derivative of the F-35 radar and almost matches its capability.
"When these shorter range radars use these missile it is not in a BVR capacity."
Baloney.
Finally, the Indian and Chinese military's are light years behind our capability. We could send them all our equipment tomorrow, and they'd still be lightyears behind. Their capabilities are limited by cultural differences as much as anything else. As an anecdote, when flying against Japanese F-15 pilots, one of the first things you notice is their lack of ability to accept criticism. Their debriefs rarely include discussions of weakness, because they don't want to offend each other. As a result, their debriefs are very short. The same is true with the Saudis and Egyptians. Similarly, the Chinese and Indian Air Forces are not trained to operate autonomously or to debrief weaknesses. That is why they can operate some of the newest Russian equipment available, but never exploit its capabilities beyond the level of a second generation fighter. The Indian/U.S. exercise you mentioned earlier is currently underway (so I'm not sure where you heard of any results). The Indian's intent is to learn and duplicate how we train. I don't think they are capable.