ASB, but ...
First probe to see the farside (Luna 3?), discovers the remains of a massive Alien base (abandoned). The US and USSR drop military spending massively, and put most of that money into space research. Clearly 3 men for a few days isn't going to cut the mustard for this, so massive LEO infrastructure is built with fuel farms, and tugs going back and forth. The USSR has a massive head start, as they classified the Luna images, so only they know for a while what's there. OTOH, their rockets are limited to Lox/kerosene or hypergolics, while the US, once they get going, have LH2/Lox, which is hugely better. By 1969, both bloks have bases on Farside investigating different parts of the alien base. While the tech is massively advanced over us, there are things that are understandable, and neither side dares lets the other side make discoveries without them.
Since the existing rocket infrastructure is vastly inadequate, massive amounts of money are poured into big dumb boosters, reusable shuttles, etc.
By the '70s, each side has a 'cheap' method to orbit, and there's infrastructure in LEO and Lunar orbit. Stations use tethers to launch and retrieve tugs/shuttles, which drops the delta-v required.
Each side by '75 has a dozen men permanently in their base. By '80, there's 100 people on the moon, there are hydroponic gardens to grow food locally and reduce the logistics needed, and oxygen (and metal) is mined from the regolith.
With oxygen and food no longer taking up the bulk of the re-supply, the stream of instruments and people increases.
'82 the first moon baby is born (not intentional, but when the mother realized she was pregnant, she hid the fact until it was too late to take her back to Earth).
By '85, massive regular flights are going to LEO, and antipodal commercial flights are also a regular thing (based on the earth-LEO shuttles). The first significant results from the Alien base are starting to arrive, and the importance of the base is inarguable.
The bases are now big enough that many of the residents are scientists and techs and maintenance people, and farmers/hydroponicists, with 'astronauts/cosmonauts' being a distinct minority. Population 1000.
By '90, the population is pushing 10,000 with Western Europe and Japan having their own sub-bases connected to the US one, and a small Chinese attached to the USSR one. Also, base has expanded, and is now split into multiple parts - residences, farms, research labs, military base, etc., etc. And an increasing civilian and purely scientific effort (on the US side) has led to the creation of Goddard University.
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Hmmm... OK, no this isn't going to work, is it? Because it might lead to an independent nation, but not a US state....
Oh well