If they want to copy your 5 year old missile, their current missiles are probably even worse.
If you have the capability to clone an air-to-air missile, you have the capability to evaluate its performance to confirm that it's an improvement over your current designs.
unless that particular model was a certified crappy missile that could not work in any way and that was a plot the have them lose time there is no reason to give them an old missile.
If you have reasonable certainty there won't be a hot war, winning a cold war is a matter of having a stronger economy and better logistics. You can leak older missile tech to the enemy, and they can work on reverse engineering it, while you install cutting-edge countermeasures on all mobile targets. They wasted a bunch of time and money catching up to where you were, and you're already 50 miles down the road. Then maybe you have a proxy war to show off your new tech. You keep the war cold by convincing the enemy that if they start something for real, they will not only lose, but also suffer international humiliation and disdain.
It also keeps your rival in the game, so they don't give up. A cold war is good for the military-industrial complexes on both sides. Peace and consumer trade cuts in to the profit margins. Can't have people choosing butter over guns.
I would love to find out that there is video evidence on tape of this actually occurring. I'm imagining that there were a group of people that planned this, and were sitting in a room watching it (with bowls of popcorn) laughing at them doing it. That means that group of people are also very good at keeping secrets. It's just farcical enough to possibly be true, until you remember actual true stories like the underwater voice line recorder with "Made In The USA" stamped on it that is now on display in the Kremlin (or wherever).
Might there be an easier answer? Perhaps the West was using a risk-based approach to security, where a missile built with aging technology just isn't as dangerous if stole as a current one.
I'm sure it's just my naive reading, but this seems like it would produce a functionally identical outcome without requiring a tinfoil hat or deliberate staking of bait for an adversary.
I think the West wanted the missile to be taken. Handing over old missile technology is pretty ingenious. If your adversary actually pulls it off and copies it, they will have a 5 year old missile. Having an old missile on your warplane, while flying against the latest tech was a death sentence. This was different than say, nuclear technology, where having a 5 year old model was still extremely dangerous.
Also, being handed old tech sucks up all your engineering resources. An analogy would be reading vs writing code. Reading code is x10 more difficult, fraught with misconceptions and inability to expand the tech.