The biggest problem with BVR used to be target ID. You don't want to shoot friendlies. The first days of Desert Storm there were hundreds of friendlies flying around and only a couple of bandits. Shooting without a proper ID would almost certainly result in friendly fire.
I have been out of the "pointy end" for over 15 years but with all the fancy new toys on jets these days target ID should be much easier now.
The incident with the Blackhawks was caused by the helos not squawking the correct codes. They were ID'd by AWACS as bad guys and the F-15's were cleared to fire BVR. Because it was a low risk bogey, the eagle guys decided to VID just in case, but they mis- ID'd them. Lots of tragedy that day.
Mig 21's great dogfighters? From an energy maneuverability point of view not really. From a real world point of view they are awesome, especially if equipped with modern Helmet mounted sights and high off boresight missiles. Mig 21's are tiny tiny little airplanes. When they point their nose at you they disappear. It is hard to fight somebody you cannot see. By contrast the big Sukhou's, F-15's, etc, are huge. They should never dogfight as they will be the first to die because EVERYBODY sees them visually from a long way off.
When you fly towards a big furball of airplanes, you start seeing big fighters at 10-15 miles. As you get to 7-8 miles you can start seeing F-18 sized fighters. 5-6 miles F-16's come into view. If you are lucky you might see a Mig21 wing flash at 3-4 miles, but if they point their nose at you they dissappear even at 2 miles. If you are lucky you will re-aquire them before they end up at your 6 oclock.