One thing before I give a full answer

The civil war was from 190. 220 is when Emperor Xian abdicated as Han Emperor to Cao Pi who founded the Wei dynasty which started over the next few years two other powers declaring themselves Son of Heaven. The civil war started two decades before when the frontier general Dong Zhuo seized control of the capital and others rose in revolt.

I can only answer why the three kingdoms became so popular, not why others did not get the same fame or the Five Dynasty period records. Don't worry, I am not going to claim the three kingdoms had better tales, better characters or anything like that.

Hope this helps

The Three Kingdoms History

The records are indeed, for the most part, good. There are things missing, biases in the text but Wei and Wu's history work gave the Shu then Jin scholar Chen Shou a strong platform. Alas, Shu never got such a project going, bar a local history one that went so bad Liu Bei put on a play about it, so their records are poor but within the political circumstances he was in, the resources he had and the biases of his time, background, Chen Shou is well regarded for the quality of his work. The Liu Song scholar Pei Songzhi supplemented it with commentary and annotations from other works that were around including local histories that had started in the three kingdoms and mystical inclined works.

However, while the SGZ is well regarded, it is not the basis for the three kingdoms popularity. The three kingdoms a lot of people think of is the fictional version credited by Luo Guanzhong much later.

Post three kingdoms

It is true that the Jin dynasty unity lasted not very long but that in itself doesn't rule out the fame of the civil war that led to it. We do not know how a united, stable Jin dynasty of a century or two or more would have shaped the view of the civil war that preceded.

Inevitably there were always tales, some based around the great deeds and the big personalities, some regions trying to tie themselves to their local heroes as local histories grew, the romanticism of poems like Cao Zhi's, there was worship of local notables and folktales spreading things like the and the folktales of the Shishuo xinyu (which also covered other eras) or others. None of this means this would have lasted over time if not for other things.

After a brief spell of peace, Jin's infighting and collapse against forces from outside China, led to legitimacy debate, was it Han-Wei-Jin with the mandate or had the Han continued with Liu Bei's kingdom in the west of Shu-Han would help shape narratives to follow. When unification would keep being short-lived, nostalgia turned to the days of the Han. Seen as 400 years of peace, stability, prosperity, of conquests abroad and protecting the borders, it became a beacon while the Yuan dynasty would attach themselves to the Han.

That Liu Bei and his kingdom argued they were a continuation of the Han, that Liu Bei was a sandal weaver who fought across the land and rose to become an Emperor, a man famed for kindness constantly as the underdog. His close friendships with Guan Yu and Zhang Fei, that the honourable Guan Yu was one of the three kingdom figures worshipped in Jing, that his chief minister Zhuge Liang was worshipped in Yi straight after he died. It all played into making them popular figures, usually against the powerful warlord Cao Cao.

Useful inspiration for small kingdoms against larger ones, giving the connection to the past, the underdog and if you were writing a play, an opera or the like, or simply telling stories at the market, you had an established audience. As religions tapped into Guan Yu worship to build their popularity, so did storytellers with works like Sanguo Zhi Pinghua or Huā Guān Suǒ zhuán about an entirely fictional son of Guan Yu (who over the centuries ascended into deity status).

While some of the tales of this era lived on, they are not why the popularity in themselves but what followed

Novel

The Romance of the Three Kingdoms by Luo Guanzhong came out in the 14th century and was revised by Mao Zonggang in the mid-seventeenth century. It took many of the tales like Guan Yu's thousand li march or fictional figures like Zhou Cang, it took popular subjects and turned into a 120 chapter epic. With duels, grand strategy, heavens involvement, two grand romances, morality, poetry and all sorts of themes

It tapped into existing audiences, existing works and created a novel that is considered one of the great literary works of China. It's popularity spread not just in China but across Asia with, according to Charles Brewitt-Taylor who translated the novel to English, a Korean saying "You can discuss life after reading Romance of Three Kingdoms." and translated into other languages.

Much of the way the three kingdoms is viewed now is through that, be it fictional figures like Diao Chan, the way the wars were fought with its duels, weaponry like Guandao that did not exist at the time, who did what where (Zhuge Liang at Chibi), personalities, who was Wei's main threat (Shu becomes the big rival, not Wu). It isn't the recorded history of the era that inspires games and TV adaptions but the romance of the three kingdoms, the work of fiction that has spread further then the history.

Modern Day

If you're creating a game, a TV drama and the like, an existing audience might be something you look for. The three kingdoms have that already, you have a large potential audience across the East thanks to the novel with plays, TV shows, games.

In the west, game series like Koei-Temco's hack and slash Dynasty Warriors that were made in Japan came over here starting in 2000, very based on the novel rather than the history, and their strategy franchise Romance of the Three Kingdoms. People played that and got drawn into the era, some reading the novel. Again, creating a smaller then the east but existing audience for games set in that era.

A game maker needs to have heard of the era and be interested to come up with the idea, an era with a classic novel and with already existing modern media about it will have more chance for that sort of reach. The novel that became a classic then plays, games, films and TV shows has created plenty of chances for the makers of a game to have heard of the era and an existing market in the east and more so in the east to pitch the product to.