Yes they are still relevant but maybe redefined. Social media has replaced some of the interaction and organization of the traditional bike club. In order to circumvent some of the legal problems created by our litigious society many bike clubs has assumed a more coalition form with only a few people doing most of the work. But even bike clubs with regular meetings can do that simply by having a signed waver by their members. And yes I know that doesn't always work but in most cases it does. Still Facebook, Google groups or Yahoo Groups even Strava clubs a bike club can be redefined. But what they also do is provide more support of an LBS. Most bike shops are willing to give a discount to local bike clubs almost like advertising. If the bike shop treats a few members of one of these social media bike clubs nice or gives them a discount then more than likely more of the club will use that bike shop.
So I guess there are now two different forms of bike club, formal and informal. The formal clubs still have a relevance because they are normally the feeder group to our racing cyclists. They do more for charity, they put on more organized and advertised rides and they more often have some political representation at least locally. The informal clubs often feed into the formal ones. Still even with social media or informal club someone has to put together the route or even have a ride leader and perhaps someone to ride sweep. So in effect it is still a bike club.