The issue with a Lunar base is that it probably would not have been sustainable even if it had been possible with the technology of the time. Apollo was on the bleeding edge of what was feasible in the J-class ‘extended duration’ missions (16 through 18) and even those lasted less than two weeks. There was essentially no recycling of consumables (air was ‘scrubbed’ of CO2 with the use of expendable lithium hydroxide filters, liquid wastes were ejected into space with no effort to reclaim water, food was all prepackaged), and all power was generated by fuel cells and stored in batteries of around 1.2 kWh capacity. Every place on the Lunar surface is in shadow for at least two weeks at a time, so even if compact lightweight solar panels were available they wouldn’t have been adequate to sustain an occupied habitat without sending a shit-ton of batteries, and I’m sure children of the era recall just how poorly ‘rechargeable’ batteries of the era fared with repeated deep cycling. There were small nuclear fission reactors and radioisotope thermoelectric generators (RTG) available but not adequately reliable enough and/or insufficient output to sustain even a small crew.
There were, of course, problems with the fine, abrasive, electrostatically-charged dust of the Lunar regolith, problems with long duration thermal management (more in rejecting waste heat rather than staying warm), and the still-unresolved issues with astronaut health with a long-duration mission in significantly reduced gravity which may pose a fundamental restriction for habitation on the Moon (or Mars). And while there is water ice on the Lunar surface it is literally rock-hard and mixed with lunar dust to form a concrete-like substance which isn’t easily extracted or filtered. There was essentially no in-situ resource utilisation (ISRU) at the time and even now such technologies are really in a very technically immature state, so relying on extracted water or other materials would not have been viable, necessitating a regular supply of consumables and all necessary hardware from Earth to the habitat, which is a logistical nightmare.
Crewed space habitation beyond Low Earth Orbit (LEO) will require extensive automation which is only possible by modern computing and autonomous control systems, and will require the maturation of ISRU technologies to a Technology Readiness Level (TRL) of 8 or better. I question that it even makes sense to establish bases on a planetary surface versus solar orbiting habitats where spin-simulated gravity can produce adequate acceleration for long term physiological health and other environmental conditions can be controlled, but that also requires access to massive quantities of space resources (presumably from Near Earth Asteroids versus hauling up material out of a gravity field) and the ability to process them into usable materials, products, and consumables.
As for the o.p., no, I don’t think a successful Soviet post-Apollo 11 Lunar crewed mission would have made that much difference. After ‘losing’ the ‘race’ to put people on the Moon, the Soviets reframed to focus on LEO space habitation and advanced that capability while the United States took a nearly decade-long hiatus from crewed space exploration (save for the all too short Skylab program and the Apollo-Soyuz Test Program flight) but made enormous strides toward uncrewed exploration of other planets including Viking to the surface of Mars and the Voyager probes to the outer planets. The expanding US involvement in the Viet Nam conflict took massive budgetary expansion and aerospace resources, and the energy crises of the ‘Seventies curtailed government spending while the always-moribund Soviet economy constrained the ability of the USSR to engage in a program beyond LEO with no real strategic or military benefit. Note, to, that it wasn’t just the issues with the development of the N-1 rocket but also the lack of a spacecraft capable of the duration of a Lunar mission and reentry from a Lunar trajectory that prevented the Soviets from sending cosmonauts to the Moon.
The idea of an extended competition in the Lunar domain or beyond between ‘Great Powers’ is romantic and has inspired a lot of alt-history fictioneering but isn’t really very plausible without enormous technological leaps in computing, propulsion, space habitation, and ISRU that simply didn’t occur and (except for computing) would make such an effort only marginally viable today.
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