The established narrative of the Space Race began with JFK's 1961 speech to put a man on the moon. In 1969, this goal was achieved by the American space program. Did the Soviets "accept the challenge" of also putting a man on the moon or were the priorities of their space program directed elsewhere? What did the Soviet narrative of space exploration look like in the 1960s in response to the JFK speech and then with the moon landing? What did it look like for the decades after the Americans successfully landed on the moon? Did it focus more on the Soviet plan to construct a space station or send missions to, let's say Venus? Did other countries (China, India, etc) offer a counter-narrative to the lunar-centric Space Race?
First off, you may enjoy my answer regarding why "the end of the decade" was set as the US goal
and my answer why the US didn't go on to Mars, even though that was one of the theoretical NASA plans
here.
If you zoom out a bit, you can think of the Space Race as milestone-chasing. When the USSR was hitting space milestones, the general thought in the US was "they're beating us again". When Gagarin went into space, one of the New York Times articles stated
The Soviet Union announced today it had won the race to put a man into space.
So that was definitely considered one of the "races" at the time. This ABC News special also gives a good notion of the national mood.
As mentioned in my first answer above, von Braun specifically outlined some possible space milestones that were potential goals, like "first radio transmitter on the moon". von Braun thought the US had an "excellent" chance of winning a race to the moon because the Soviets did not yet have a rocket capable of such a mission, so the US could focus on the moon and leapfrog to their own milestone.
This milestone wasn't necessarily going to be the "end" -- it was a way to target development in an encouraging way (both for morale purposes and for political purposes). The failure of the N1 rocket (spoken more of by /u/kieslowskifan here) essentially led to Russia "dropping out" of the tit-for-tat aspect of the race. Had the Soviets followed up with their own moon landing shortly after the Americans did, it is quite possible a Mars mission would have gone forward to "keep the race" going, but since the Soviets ended up with a clear change in focus, there wasn't any more definite "race" milestones to attack where both countries were aiming for the same goal.
So to reiterate: you could think of the Space Race as a series of races, where the last accomplishment -- and the most technical one -- was won by the US. The milestones have now been wrapped into a grand race, hence the current narrative.
The Soviets were definitely trying to reach the moon, but because their effort petered out, the Apollo landing was under-reported in the press. I'd say the accurate notion is them pretending there was no race at all. /u/Dicranurus has an excellent answer here which goes more in depth on the matter. To quote one part:
Publically, to limit embarrassment, the Soviet Union proclaimed to only be interested in an orbital space station.
The only other country I know of at that time who had any notion of "joining in" the race would be the UK, who did technically have a headstart an early launch of a V1 rocket, but they never publically announced anything that would constitute milestone-chasing.
While China started with space in the 1950s, China's own manned program didn't get kicked off until 1967; they were behind enough that any milestones in that time period have been national goals, rather than based on any sense of competition. (Although, to briefly hit a current event: in 2019 the Chang’e-4 probe made the first soft landing on the far side of the moon from any country.)
One element often missing from the popular form of the narrative is a sense of cooperation. After the John Glenn launch, Kruschev wrote directly to JFK:
If our countries pooled their efforts—scientific, technical, and material—to master the universe, this would be very beneficial for the advance of science and would be joyfully acclaimed by all peoples who would like to see scientific achievements benefit man and not be used for "Cold War" purposes and the arms race.
At the United Nations in 1963, JFK suggested cooperation with the Soviets on a moon landing, but he died a few weeks later.
The first actual joint mission was in 1975 with the Apollo-Soyuz Test Project. The Mir/Shuttle partnership from the 90s (which I wrote about more here) constituted "Phase 1" of a partnership between countries, leading to "Phase 2", the International Space Station, where many more countries (incuding China) were able to participate and be at "the forefront" of milestones. So if there's anything like an "alternative narrative" at all, it's the idea of working for scientific cooperation, and not bothering with the "racing" aspect.