(or "How Does He Get The Costume Back In The Ring, Then?")
When looking at the classic era of the Flash represented in 'Showcase Presents The Flash', you really are looking at one of the best cases of intentional storytelling engine design of the Silver Age. Julius Schwartz really did try his best, given the way comics were being written at the time, to actually think of an entire status quo for Barry Allen that would lend itself to numerous stories (which makes sense, given that changes to the status quo were few and far between in that era.) He gave Barry a romantic interest, a locale, and a job that lent itself to being a super-hero.
It's that last one that's worth another look, particularly in light of the modern era of comics. Right now, it's getting difficult to find a superhero that actually has a "secret identity"; over on the Marvel end, Spider-Man, Iron Man, the Hulk, the Fantastic Four, the X-Men and Captain America are all publicly known to be super-heroes (or at least, their corpses are publicly remembered to be super-corpses.) Many people in the industry are claiming that secret identities are a vestigial concept from an earlier time, like kid sidekicks and the Comics Code. So, is it true? Why would a hero have a secret identity?
Obviously, it's to protect their loved ones. (No, it's really not. That might be why they don't blab it to your average man on the street, but I'm pretty sure my mom could keep a secret if I asked.) That's always the reason they give, but the real reason is that a secret identity can be very helpful to a writer. Look at Barry Allen. He's not just a super-hero, he's also a police scientist. That means that not only can he get involved in a story the way any superhero can get involved in a story ("What's that? A cry of distress?"), he can also get involved in a story the way any police officer can get involved in a story. Given cops are expected to find out about crimes, it provides any number of angles for a writer to help start off a Flash comic without seeming contrived. (The She-Hulk, a lawyer, and Superman, as a reporter, also have similar secret identities that help them get involved in criminal situations. Batman, as a millionaire playboy, always has to have society friends casually mention a crime to him, though.)
But more than that, the secret identity becomes a separate sub-genre of stories in and of itself; like any secret, it takes work to protect a hero's true identity from discovery, both from the public and from friends and enemies. Barry Allen has to keep reporter Iris West off the trail of his dual identity (and notice, by the way, that Iris has a job that doubles as an additional entry point into stories for the writer; she frequently mentions to her boyfriend a story that he decides to follow up on as the Flash.) In this regard, Iris is just one link in a long chain of nosy friends, family and well-wishers; from Lois Lane to J. Jonah Jameson, people are always trying to find out the hero's biggest secret. (Ironically, the current Star-Spangled Kid got her start by being the nosy kid snooping on the super-hero.) It's always good for a story, and that's an advantage to any storytelling engine.
So is the secret identity "outdated"? Doubtful. Certainly, as modern surveillance technology improves, it gets increasingly implausible that someone could hide their identity that easily (although Barry Allen always had the advantage of super-speed.) But it's a useful storytelling technique, which means it'll probably never go away. In fact, Barry Allen had his secret identity magically re-concealed a few years back, after publicly revealing it. In a universe of comics, "implausible" gets stretched a bit further than it does in the real world.