Maybe not strictly an everyman in EVERY sense, but I always saw Pete as more relateable and closer to us normal schmoes than other heroes. Sure he's a genius, but he's not Tony Stark living in a mansion, slinging back martinis, he was troubled by "normal people problems". At his youth it was money problems, dealing with school, girl troubles etc. As he grew up it was still money problems etc.
I think the whole thing is that he's closer to us in certain aspects rather than huge heroes:
- He's a super hero, but for years he was dogged on and never looked up to universally like Captain America or Superman are
- He had the dual identity problems, but didn't have the luxury of riches or a realistic, but successful job (Bats, Tony, Daredevil, Supes) to finance his adventures. (While he could pay for things, he still had to budget and be careful)
I dunno, I think saying he's an everyman isn't really the best way to describe Pete now, but I think the whole thing is that even wrapped up in the fantastical he's still or was intended to be closer to us folks than some of the others heroes.