That’s fine. As long as things are moving forward, or at least it feels that way, it’s not a big deal. In theory, it’s possible to talk on the phone, but I’m not convinced that would go better or faster (in fact, it would likely be the opposite).
Regarding navigation, @Benjamin_D has already provided the solution.
As for the first point, as mentioned above, what do you mean by “see”? Do you mean it shouldn’t be accessible or visible at all, or just that it shouldn’t appear in the group’s “activity” tab? (It would still be visible and accessible elsewhere).
If we use the analogy of public activity being broadcast on TV, do you just want it not to appear in the TV schedule? Or do you want it not to be broadcast at all, or for access to the channel to be impossible?
What is public is visible. Either it shouldn’t be visible (making it private, which also limits the number of potential participants and thus the possible discussion), or it should be made inaccessible (no public part at all. In that case, it’s clear there won’t be any discussion “outside the group” if that’s the only thing they have access to). There is still the question of private messages to consider. In the worst case, you can also create separate forums for each group of people, who would then be completely isolated, with potentially different settings/configurations. That is one solution/option.