DC has plenty of underdog stories as well. For example, anti-hero Catwoman starts out as a simple cat burglar, trying to eke out a living amidst the madness of the city of Gotham. Barry Allen was just a police detective when, much like Peter Parker, a laboratory accident granted him amazing superpowers. Like Spider-Man, Allen's superhero persona, the Flash, often acts as a grounding, humanizing figure to his more powerful superhero colleagues.
Furthermore, Marvel's roster may contain some average Joes, but it's also filled with extremely impressive figures that normal people can't relate to. Just look at how many rich geniuses there are, like Tony Stark, Bruce Banner, Reed Richards, and even Victor Von Doom. Or look at the Asgardian royalty, Thor and Loki, who are basically gods. And even when Marvel has a "normal" hero, it throws them into such outlandish circumstances, like the aforementioned multiverse plot, that the audience can no longer relate to them. It becomes a fantasy story with a different paint job.
And at that point, if the story becomes so fantastic, why bother grounding your heroes in reality in the first place? Fantasy isn't a bad thing; audiences crave escapism. But if an audience wants to escape the real world for a bit, why would you remind them of it with an ordinary main character in the ordinary world? This is where DC's godlike, larger-than-life heroes comes in, and fills a better role than those of Marvel.