I would suggest you check out Discord, which works well for both the team and individual use cases in one user interface. The way they accomplish this is by having "Friends" (a familiar analogy for anyone who has used chat applications from the 2000s and current social media) as well as "Servers", which are akin to teams.

Joining servers is very easy, creating servers - they are perhaps inaccurately named - is instantaneous, and each server can contain groups, voice and text channels. Adding friends is easy, and from the friends menu it's easy to message or begin a call with anyone and create ad hoc group chats and calls.

Perhaps your belief that they are entirely different problems and necessitate different solutions is based in the fact that they the market has thus far assumed that to be true.