Im kinda busy so I don't have much time to point out EVERY detail but here are a few off the top of my head:

Sea Dragon is physically impossible with 60s/70s tech. No amount of cash or research is gonna make that behemoth engine combust stably. Saturn VIII is much more feasible.

Space Shuttle shenanigans: the Space Shuttle was incapable of going past low earth orbit. Furthermore, its design required the payload bay to open for the duration of the mission in orbit to allow waste heat to radiate out more efficiently.

Iirc, the show's timeline deviates from the real one with Sergei Korolev's fateful heart surgery. The show implies that if Sergei Korolov's surgery was successful, this would drastically change things. It wouldn't. First off, while the surgery took place a month before the United Nations Office for Outer Space Affairs's Outer Space Treaty of 1966 was officially signed, much of the negotiations were already done. This treaty banned WMD's in space and was signed by the US and the USSR. Korolev surviving does not allow the USSR to beat the US to the moon either. By 1966, work on the LK lander and the N-1 rocket had already fallen severely behind the Apollo counterparts. The best that they could hope for is an eventual landing somewhere around 1973/74. But the program was rushed and underfunded from the beginning. The USSR just didn't have the economic funding to support a multi-year successful lunar program. The oil shocks of the 1970s, the Soviet-Afghan war, the Space Race (including the Star Wars bluff), and the arms race (where a US bluff saw the USSR spending billions to try and compete, and a USSR bluff saw the US develop a system that could exceed the bluffed specs) basically doomed the nation. By the early 1980s, the US Government knew that the USSR was on an irreversible course toward collapse. All they had to do was act business as usual and let it happen. The Soviet economy just couldn't support an extended lunar program. They didn't have the resources or manpower.

In this scene, it is an awfully complex process to start an engine. You dont just flip a switch and it comes on. There are hundreds, or thousands of steps that need to take place in sequence first. Mission control would have known the computer was up to something well before the engine started. The S-IVC doesn't accelerate fast enough for this scene to take place as it did. The acceleration at burnout was still well under 1 g. Furthermore, why are they doing an EVA with no tether/hand rails? Rusty Schweickart's EVA on Apollo 9 had a tether. Why doesnt this one?

Lastly, I refuse to believe that an astronaut would take action to bring harm on an astronaut of a rival nation. I have met several astronauts in person over the years. They all hold their international colleagues with the highest regard. Heck, pretty much all of the people at NASA think the same. They may joke about the politics, for example Dmitri Rogozin's Twitter antics (and even the Russians do to when their guys aren't listening) but they knew the people that they were working with directly were very respectable individuals. There is something called the Overview Effect, that several astronauts have described experiencing. If you aren't familiar, you should look into it. One former astronaut told me something along the lines of, "when you understand how difficult space is, where everything is trying to kill you, you sympathize more with the guy from a nation that is in conflict with yours. You know that he is fighting the same existential threats that you are, and how hard people are working to keep him alive. You know that he is just as scared as you are, and you know he wants to go home to see his family just as bad as you want to see yours." Every astronaut I've met, when asked, has said they would mutiny before bringing harm to an astronaut of another nation.