I don't think they'd have any particular problem with it - this isn't really a new thing in comics, just a rare one. Look at Ralph and Sue Dibny - true, they're a couple, not a family, but they fit perfectly well into the Silver Age, and he doesn't even have a secret identity. And Superboy having a loving mother and father who shared his secret was an important element of his stories. Not to mention, of course, the 'World's Finest' Batman/Superman team-up - they may not have been family, per se, but they and their respective supporting casts hung out together so often that they must have been pretty chummy with each other. Robin and Jimmy Olsen would help each other out, Lois Lane and Vicky Vale probably met a few times - it was like a sort of extended family. Families and traditional superheroics can go together - it simply means expanding the group of people you trust to keep your secret.