^ From what I can tell, the main thing about teams is that it forces team members to work together on a more regular basis than any other type of activity, except for maybe dramas. For example, Yukirin wasn't close to her French Kiss fellow members because they were in Team A and she was in Team B. Even being a senbatsu regular doesn't guarantee interactions, as they tend to do their promotional activities in smaller groups and pairs that are decided by available schedule of each girl. But teams have to go through the long process of learning a stage together, and then rehearsing and performing it every few days. That kind of long term interaction inevitably forges bonds and is the platform for many friendships.
That's why Mayu was crying so hard during the original shuffle when Rabutan got transferred to A, and you can see that they aren't as close anymore after the fact. It's similar to when girls were graduating from MM but staying in H!P. There's still a huge gap that opens up simply because they aren't working together as much anymore.
The other important thing about teams is the stages. A lot of stage songs are written specifically for the members in a stage unit, or the general temperament of the team, so having a particular combination of members in a team can make a particular kind of stage song possible. For example, K6's Uhho Uhhoho. No3b formed with the roster it has because they already saw that Kojiharu, Minegishi, and Takamina could work together really well in A4's Junai no Crescendo and H2's Bye Bye Bye.
Similarly, the general temperament of a team can define the image of the members in it. Team K was the house of crazy wild childs, Team A was meant to be more polished traditional idols, and Team B were the quirky girl-next-door types. After the first shuffle, the members had to change themselves to suit the team's image.
Of course, you're not wrong, because Teams don't play that big a role anymore, and it's completely possible to enjoy the fandom without caring about them. Unless you care about pairings, because my first paragraph still stands.
If you really want to know why teams matter, I'd recommend watching To Be Continued, as the girls themselves talk about how the shuffle affected them and how they feel about it. Kasai, Minegishi, and Itano's interviews, specifically.