Personally, I'd worry less about the headiness of these books than what seem to be common themes of characters whose defining characteristics are innate - birth as a demigod, half-wizard or having a high midichlorian count.
Does anyone still write about 'normal' people doing great things?
There are plenty of non-innate ways to make the protagonist remarkable in some way, and most can be hand-waved away as not important to the plot. Epic heroes are easy to follow.
The magical origin story simply does not interfere with the adolescent empowerment fantasy. You, too, could be the secret bastard of an immortal, kid! That makes you special and important and worthy of respect! Tonight's homework: being awesome. Tonight's chores: save the whole freaking world.
In those stories, the protagonist of the story is proxy for the hero in you, and it saves the author a whole lot of character development. If the main character is instead some sort of average Joe Schmoe, the author has to figure out a way to make the reader care about him, whether that be sympathy, pity, hatred, or whatever.
There's good reason why some fiction stories seems to follow a formula. It's because the formula works. Define hero. Add conflict. Introduce the very likely possibility of defeat. Wind up to an epic confrontation, where the hero is ultimately victorious. Resolve subplots and romantic tension. The end.
If you want to read about normal people doing great things, read a nonfiction biography. In fiction, there are no normal people, because every author has the mystical superpower of invoking the suspension of disbelief by any means necessary.
The stories are set in fantastical worlds to make them more interesting to read, but the issues they address are mostly "normal" young adult issues like alienation, conformity, identity, family strife, censorship, oppresion, racism, bigotry, etc.
It seems like you're trying to say that you want to read stories about people who are 'normal', but that doesn't really mean anything. Are you just looking for books that don't have any element of fantasy to them?
Is this character conspicuous because things worked out for him?
Or did things work out for this character because he is conspicuous in some way?
Game of Thrones has a lot of the first, virtually no teen book does. They are imbued with something special and everything works out because of their specialness. Sometimes multiple other characters join the cause sacrificing or putting on hold their lives because this person is a prince(ss).
With all this dystopian teen drama going around, I wonder how many SF authors are out there kicking themselves for not writing a romantic triangle into their story.
It's a standard plot: ordinary person living an ordinary life feels constrained by that life; boom thing happens, causing ordinary person to go on a journey. Character development happens on the journey.
I'm kind of surprised that you can't think of any book like this.
edit: I mean, Harry Potter fits the bill you outlined, but is explicitly called out by the parent post as a half wizard.
I read James Patterson's "The Beach House" on a pair of short plane flights this past summer. It's a commercial/pulp thriller. The MC doesn't acquire any superpowers during the story except for becoming a badass willing to take significant risks when his life and that of his friends is on the line.
Edit to add: Jack Ryan, especially in the early Tom Clancy novels.
In any story worth reading the main character won't stay "normal" for the duration. The protagonist's change through the course of the story is part of the point of telling a story.
Even after that, I'd argue that Frodo is only made interested through his posession of the one ring, not anything about himself. Sam is probably the only really interesting 'normal guy' in the story, and its only because he gets to tag along with the godly guys.
If you dig deep enough, every memorable character is a snowflake.
Even Walter Mitty, possibly the most average man in literature that I can think of, was exceptional in his imagination, and it is that exception which makes his story compelling.
Maybe Catch-22 is another example? Yossarian is a pretty average soldier, from what I recall.