When it comes to costumed superheroes it is always about the costumes and the people inhabiting them, never about just one or the other. The costumes of Batman, Superman and Wonder Woman had already undergone significant revision from their original versions by the beginning of the Silver Age, yet it is such revised versions that are largely regarded today as their "classic" looks. This happened because the essential lexicons of the original versions of the costumes were carried over. Ridiculous as it may seem on the face of it, Batman must have pointy ears on his cowl batwing shrouds to be Batman, Superman must have a big red S on his chest and a billowing red cape to be Superman, and Wonder Woman must have the red bodice with gold appointments and blue shorts with white stars on them to be Wonder Woman. These aren't just decorations, these say who these characters are. Remove them, and the costumes are rendered mute.
It has to do with continuity. The original version of the Flash effectively disappeared from the comics racks at the start of the 1950s. The idea of the Flash was revived five years later, but as a new character with a new costume on what would eventually come to be understood as an entirely different planet Earth. And it proved the same for such characters as The Atom, Hawkman, and Green Lantern. This was re-imagining on an enormous scale, but years-long gaps between versions of these characters was enough to bring up an entirely new generation of readers that had no ready awareness of or investment in the originals.