Disagree completely. Wally's been a superhero since like age 14. Barry Allen had a normal life, a schooling, a job, hobbies, and interests. You're misusing the word "everyman".
With Barry Allen superspeed really is a gift and wish fulfillment.
You're saying that getting the powers of your favorite super hero and getting to become his sidekick isn't wish fulfillment? Also, I'd contend that Wally is more of an everyman than Barry. Barry is a police forensic scientist, which thanks to shows like CSI carries about it a mystique that people are fascinated with like it's some kind of magic. He was already a "crime fighter" before becoming a super hero, so becoming a superhero on top of already having a cool police job is somewhat redundant.
With Wally, on the other hand, he's often out of work and has often had to change professions. It doesn't matter that he's had superpowers since he was a little kid, because it hasn't kept him from facing the same struggles that we all do in life. In a lot of ways, Wally West is the Peter Parker of the DC Universe. He's a likeable, funny guy who's struggling to get by like we all are, while also fighting crime as a superhero-- while also carrying the weight of a legacy on his shoulders, which is something that very few heroes have to deal with.
It's not a source of conflict with him. With Wally, the legacy and superspeed have their drawbacks and burdens. That's the primary difference between the two, along with Wally having more of a motormouth, not any sort of "everyman" quality.
Stories are about conflict. If Barry doesn't feel conflicted at all about his powers, then it's all the more reason to go with Wally since in terms of storytelling internal conflict > external conflict. Also, Wally is funnier and more talkative than Barry, which I think is an asset to his character. Barry is a lot more straightforward in his personality, which IMO would make for a less interesting character on screen.
Personally, I don't think it's really a problem as far as a set of movies goes. A first movie, with Barry Allen, is likely to have the character arc of transition from ordinary man to hero.
Which Fantastic 4 has already proven is getting stale. With Barry Allen, it'd be the same shtick we saw in Spider-Man of "he gets powers, learns how to use them, and designs a costume, and becomes a super hero." At least with Green Lantern it has the exciting hook of him getting to go into space and be trained by an elite corp of inter galactic guardians. If they want to draw people in they need a less generic origin, and there's nothing generic about a superhero dying and his sidekick taking his place, since it's never been done in a movie before.
Look at these two loglines, and tell me which is more interesting:
A forensics scientist gets super speed powers, and decides to become a masked crime fighter to stop a crazed villain threatening his city.
A sidekick must take up the mantle of his dead Superhero mentor, and defend his city from a crazed villain.
One of these, with only a little modification sounds extremely generic, wheras the other has a unique twist that makes it high concept. High concept sells, and it gets people excited enough to go see a movie. Barry Allen's only hook is that he's already fighting crime before getting his powers, and frankly I think that's a pretty boring one.
I also have to ask, are people wanting too much soulsearching here? What's wrong with a likable protagonist, with confidence and a sense of humor, having exciting, fast paced adventures? Spider-Man is one way to do action adventure, but it's not the only way. Indiana Jones and James Bond have traditionally had more focus on plot and action than on the character being on a significant arc.
Superheroes are a great opportunity to explore the human condition, which is why they do it so often. There are superhero movies out there like Fantastic 4 which focus more on action than studying the characters, but they usually get poorer reception because they're just shallow entertainment.