This is why work friendships in my experience are quite easy, even for an introverted hermit like myself. Often common interests between people, and _tons_ of time bonding.
If i really cared about forming more friendships i'd do what you suggested - find groups that interest me and spend the time needed to form friendships. I'm also a little weird in that while i get along with a lot of people and i am (hopefully) easy to get along with, i'm generally not interested in relationships without common interests and activities. Because i don't like to socialize purely for the sake of socializing - i like building something together, or playing something together, or enjoying food together, or whatever it is i am actually interested in.. which is usually software.
[1]: Which is probably useless, as i'm generally an introvert who still has less friends than i'd prefer.
Really depends on your team, in my experience. Nothing wrong at all with being a middle-aged parent who's too busy to do things with coworkers outside of work, of course. It's a workplace, and we're here to do work, not hang out. But my friendships at work tend to happen with people who are around my age and are interested in doing fun things together outside of work too.
Maybe it depends on your definition of "friend" as well. For me, no matter how well I get along with someone at work, so long as we're only chatting with each other over lunch and coffee breaks and the occasional team event, we're just acquaintances, not "real" friends in the sense that you hang out just for fun and help each other out with things that come up in personal life.
everywhere i travelled or lived i was easily welcomed to be an active member of the local user group and from several i have friends that still remember me a decade or two later.
i don't even remember any smalltalk phase (except for the smalltalk programmers group :-). there usually was some topic or someone with a question and we launched straight into technical discussions. all i had to do was show up and participate occasionally.