Kennedy’s ‘reconsideration’ idea of trying for a joint Lunar effort was strictly because, (as I noted) he really never wanted to go there in the first place. By the time this comes up it’s becoming clear the US isn’t that far behind, (if it every actually was) the USSR and it’s not inconceivable that he can somehow ‘walk-back’ the initial rhetoric. Then he gets whacked and it’s now a martyrdom thing with NO chance anyone can stop it. Unfortunately once it’s clear we ARE going to be first and there’s nothing the Soviets can do to stop us the moment quickly dies and the cuts begin. (NASA’s budget begins falling four {4} years before the first landing)
THE thing to keep in mind is even if Kennedy lives and DOES manage to de-escalate the “race” and propose a joint mission the Soviets are likely to refuse simply because they know they will be the ‘weak-link’ in any such plan and they will no longer be able to obfuscate or hide how marginal their actual ability is. There is no ‘upside’ for them in such a deal so they’d likely politely decline and keep ‘focusing’ on Orbital operations which would leave the US to either follow suit or keep going for the Moon and frankly once the ‘pressure’ is off we’d likely cut back ourselves.
Once the Saturn-1 becomes available the Soviets have ‘lost’ the payload race and they knew it which was the reason the UR-500 (Proton) was greenlighted. But it turned out to be much harder to make operational than they thought, (not having the N1 probably would have helped) and was far less reliable than originally thought. “Worse” from their perspective was the US was developing the Saturn-V, (though again without an actual Lunar mission in less than a decade Its ‘utility’ is questionable at best) which was going to blow the UR500 out of the water. EVEN worse upgrading the Saturn-1 was going to be ridiculously easy so even without the Saturn-V the ever present ‘gorilla’ in the room of US productivity and industry had already been unleashed.
There’s just not any way to make a plausible ‘upside’ to such a cooperative effort. “Apollo-Soyuz” was an embarrassment to the USSR and any ‘joint’ lunar mission is going to be worse unless the US effort and operations are curtailed down to the same level as the USSRs. Actually I CAN see one (1) possible ‘upside’ but it requires the US leadership and NASA be rather stupid; By joining in a ‘joint’ mission the USSR can degrade the US effort to the point where they cannot possibly reach the Moon before the mid-70s to late 80s by insisting they are given important development and operational subsets of the mission… And then consistently fail to meet any of the goals or milestones and cite “more important” goals which would play well during the US technology ‘backlash’ of the late 60s and early 70s.
“Going to the Moon of course is a laudable goal but we have decided that we need to feed people/alleviate 3rd world poverty/address the problem of pollution/etc” will play VERY well worldwide around that time and the US will look rather selfish and arrogant, (not like we’ve not been THERE before but still…) for insisting the Soviets stick to the program. And if we’ve ‘agreed’ to go with them we certainly won’t find the support, resources or money to do so alone…
Randy