Addendum

Did some more reading on the AIM-4D.

The LAU-42/A launch rail for the Phantom had enough coolant for "several" firing attempts.  But not near enough to keep it cooled for long at all.

To save on the refrigerant, you didn't ready the missile until you thought you'd need it, but that means waiting 5-6 seconds for it to cool down.  You could leave it on, but you only get about two minutes total before the reservoir is depleted.  Additionally, this reservoir is only on the inboard rail, it is shared with the rail mounted to the pylon if a second missile is carried.

Because the AIM-4D is adapted for stand-alone use it has a post launch "boot" period where it has to stabilize.  It flies in a straight line for about a mile then begins to maneuver.  If the target aircraft is still within the forward facing cone of the seeker and the seeker is inside the requisite cone behind the target, then it's got a decent chance of hitting.

As I mentioned before, a fighter can put a 5g turn on and escape the slew rate of the seeker head when the missile is about a mile away.  This is also true of the AIM-9B.

In Vietnam two things conspired against the AIM-4D.

First, the rules of engagement required visual ID of the target.  This usually meant getting within five miles of the enemy and being head on to them.  None of the heat-seekers of the time could be used in this envelope.

Second, the Vietnamese were flying nimble planes armed primarily with guns.  They wanted to be in close and to stay there.  Getting a Phantom away from a MiG-17F far enough to use a Falcon and keeping him there long enough to get shot was asking a lot.

These same rules of engagement affected the radar guided AIM-7E missiles which had a nominal range of 15 miles and COULD be used head-on.  By the time visual ID was made, it was nearly always impossible to make the shot because the trigger was just the last action the crew took and not the last part of the process.  1-2 seconds could pass before the weapon was released and with 1,000 knots of closure, 5 miles is gone in a snap!

The missile that got the Air Force thinking that a Falcon would be a dandy replacement for the AIM-9B was the AIM-4G.  It was a larger version of the Falcon and was mated to the MA-1 fire control system in the F-106A.  The refrigerant tank on the Delta Dart is enough for several minutes of cooling and the missile leaves the rail running.  This is because the IR seeker, radar and computer on the plane is doing the calculations for the missile before launch; additionally the computer is flying the plane to the optimal launch point (not something that lends itself for use against fighters, by the way).

The Swedes made further developments of the Falcon for their Draken and they addressed many of the shortcomings.