First of all, they're not strictly childhood friends in the common manga/anime trope sense. They didn't live next door to each other, they didn't attend the same primary school, etc. The first should have been obvious since Seika didn't know where Kousei lived until
Chapter 2 in the current timeline. They knew each other from many years ago when Kousei created a gift for her while she was staying in a local hospital. It's been
established that he gives gifts like this on the regular. He definitely remembered creating and gifting something as specific as a doll based on an anime character, but as also previously indicated, he's
bad with names, so he doesn't quite remember the name of the girl who received the gift. Spoiler: it's Seika, so in the heat of the moment, she takes it hard and runs away, though she hard regrets doing so later. Anyway, this confusion gets sorted pretty quickly, so it's not a major impediment to the story.
As for the critical plot development, the reason Seika's in a hospital close to Kousei's neighborhood is because Seika's mother wanted her to be treated at a place near her own birthplace. That is, Seika's mother prioritized the familiar environment of her own youth over the quality of health care. However, Seika's father wanted her to be treated at a more prestigious hospital with better care. And after a few months, without seeing much improvement, Seika was relocated to a better hospital where she got better treatment and overcame her illness. However, the gift she'd received during her first hospital stay never left her memory, as well as the boy who'd created it for her, which is why she specifically chose this high school to attend (away from her current residence) in the hopes that she could reconnect with the person who helped raise her spirits while she was ill. And though it was happenstance, she did reconnect with her first love.
Yes, the story can be told in many different ways, but I wouldn't say that her specifically choosing this high school hoping to find her first crush is a horribly worse story than simply finding a crush randomly. The main romantic beats for a Japanese high school romance are still in place: the MC, while being thoughtful and honest, is not overly proactive, and the FMC is not especially aggressive and is prone to blaming the MC on occasion, such as this. But that's common enough. People in general, and grade school students in particular, are often self-absorbed and see the world from their own perspective, being quick to judge when someone else doesn't see things the same way they do. So it's natural that Seika would place special emphasis on the gift and relationship she established while she was ill, but that Kousei might not have done the same since he gives gifts to a lot of people.
But, as previously mentioned, this gets sorted pretty quickly, though it'll still take some time before they're officially dating. I went back and checked. Chapter 4 of the manga concludes with the end of
Chapter 14 of the web novel, but they don't reach a confession and become boyfriend/girlfriend until
Chapter 132. So yeah, it'll be a while.