This is very subjective and each person will define a 'friend' differently. I personally, if I don't count my wife, I can count the number of friends with just my fingers, but for me, the word friend means someone that are there for you for whatever, that even after not seeing each other for a few years, there is no awkwardness and you can pick it up from you left. For others, someone you see regularly, have a few beers are considered friends.
I'd say 90-95% of what we do is play board games together, but I know what's going on in their lives and they do mine, and I've attended their special events like birthdays and housewarmings and weddings, and helped them out when they needed it and supported them in their endeavors.
If that's not being a friend because we mostly just do a 'single activity', then I very much disagree with your definition.
I do agree about not having seen each other for years and there's no awkwardness when you get back together being a mark of friendship though. I have that with my old high school friends, and I'm sure it'd be the same with my main friends today.
This is very different from 'these are people I only play boardgames with them'. I have tennis 'friends'. These are people I only see in relation to tennis (either playing, practise or even go to watch a match with) but wouldn't get invited to their wedding or be asked to help them move for example. Sure, when we are at the club we chat a lot and are very friendly, but when I'm looking for someone to take care of my kid or lend me his car because mine broke, I won't be calling them (doesn't mean they can't turn to people I would in the future, but pretty sure for that to happen the friendship would have moved off the courts)
I don't connect with people via words. I've always felt people are primarily defined by their actions, so it is very much about the "doing" for me, and it's through doing things together that I make connections with others. Obviously, my friend with whom I rock climb isn't just my rock-climbing partner - we do other things as well, many of which are unplanned and are new experiences. But it's the possibility of playing out our spur-of-the-moment ideas, as well as the other regular hobbies, that make our friendship even remotely interesting. Without being able to do those things together (which is way, way more difficult when you have to take a plane ride to visit), neither of us would be able to glean any meaning from the other's existence.
I don't know - was I really far off in how I interpreted your comment? What is a friendship, to you?