WARNING: The following contains spoilers for Horimiya Season 1, now streaming on Funimation.
In the anime world, many non-violent series involve a handful of well-established character archetypes, many of them based on female characters in particular. Nearly every anime fan can recognize the tsundere type on sight, and the same may be true of yandere characters too, such as Himiko Toga (My Hero Academia) or Yuno Gasai (Future Diary). But what about the stock "shy girl" who can be found in the story's periphery?
Such characters can be endearing, since they are nearly always innocent in both their actions and intent, and they draw viewers in with their gentle and warm personalities. However, such characters rarely go the distance. They are often treated as a novelty instead, though a few anime series break this trend with style.
Why Can't Shy Girls Win?
If an anime series isn't focused on action and adventure, such as becoming Hokage or winning in a combat or sports tournament, then the stories are more along the lines of forming a romantic relationship, achieving self-improvement, or overcoming grief/guilt or something similar. Shy girls in anime may have these personal stakes too, involving either self-actualization or winning another character's heart, or even both at the same time. If a shy, chronically lonely girl can score a stellar boyfriend, that's a real victory. But can she get it? Maybe not.
One might wonder: why do these shy girls often fall short of their goal? Why can't the timid, bookish girl impress the male lead and win his heart? One possible reason is that the biggest stakes, such as romantic goals, are best suited for more extroverted and assertive characters, where everything is in proportion. That is, "find my soulmate" might simply be too big a goal for a shy and plain library girl, so her stakes are scaled down to something more like "make five good friends this semester" or "get into a good college" or the like. This isn't to say that shy girls don't deserve their Prince Charming, though.
Instead, anime series often complete a shy girl's character arc with her achieving quieter but satisfying goals, such as moving on from past trauma, making a new friend or even learning a new hobby or getting into college/getting her dream job. Not everyone needs to ride off into the sunset with their lover. The female lead can take care of that, while the shy girl gets her desired job or forgives a past enemy and gets an emotional release.
It's also possible that manga authors and animators expect fans to be underwhelmed by the scenario of the shy girl "winning" a harem race to see who can become the male lead's soulmate. Shy girls are popular for what they are, but such a timid and unassertive character might even feel "weak" or watered down (or simply lopsided) when paired with the male lead, while a more assertive, flamboyant girl would complement him better. Or, the shy girl's actual personality and goals might preclude her from even wanting to find her soulmate. She might feel overwhelmed by the very idea, and it would be cruel (or just inorganic) for the narrative to force her to pursue the male lead's heart anyway.
A few particular anime girls fit this trend. In Horimiya, Sakura Kono is interested in Toru Ishikawa, but Toru has his heart set on Yuki instead, and Sakura doesn't dare interfere with their budding romance. Then there's the matter of Shiori Shinomiya from The World God Only Knows, a classic harem series. Shiori gets a decent character arc, but she is miles away from being the male lead's girlfriend. In We Never Learn, the warm-hearted but quiet Fumino Furuhashi has an interest in the male lead, but in the end, she doesn't become his girlfriend. Instead, she is more interested in pursuing a college career studying astronomy.
When The Shy Girl Wins (Or Stands A Good Chance)
In The Quintessential Quintuplets, several of the Nakano sisters are racing each other to see who can win Futaro Uesugi's heart, and so far, the assertive Ichika and the shy Miku are roughly tied for the lead. Futaro will definitely marry one of the Nakano sisters when they're older, and Miku, despite her timid and quiet ways, stands a very real chance. Manga readers, meanwhile, know for sure whether she wins (spoilers). Even if Miku loses to Ichika, it's remarkable that such a shy girl has created such stiff competition for not only Ichika, but the ultra-assertive Nino.
Likewise, there are some completed series where the shy girl really does win, or ongoing series where she stands a solid chance. If the definition is stretched just a little, then Orihime Inoue from Bleach is a victorious shy girl, being much more reserved and modest than the tomboyish Rukia Kuchiki. Despite the strong Ichigo/Rukia chemistry, Orihime married Ichigo in the end (even if many fans wanted a different outcome). Sometimes, slow and steady really does win the race.