Keep Multi-NPC Conversations To A Minimum
Obviously you can't always do this-- sometimes you need a town council, or talking down an angry mob, or a trial, or some other crucial scene where you have to let the PCs try to influence multiple NPCs with multiple viewpoints at the same time.
But for all the reasons you point out, keep them to a minimum. They are stressful, and they can end up being puppet shows where the PCs just watch something happen, instead of participating in it.
Keep Active NPC Participants To A Minimum
Even if you have a village council meeting with 5 or 7 or 39 or however many members, there is rarely a reason for all of them to be speaking to one another. That doesn't happen at well run public meetings anyway, because it turns into chaos. Most formal meetings will have formal or informal rules that limit that chaos. You can emulate that by having only one major stakeholder for each position in the meeting, which will probably be limited to two or three.
Even a wild mob will have probably have a leader or someone (or a small number of someones) who take it upon themselves to shout on behalf of the crowd. If not, it's not a dialogue scene, it's a fight or a chase scene.
Involve The PCs In The Turn-Taking
This is not real life, this is as artificial as a court room drama in a movie-- you don't want every NPC to talk in between each instances of the PCs. You want, at best, one or two NPCs talking, and then the PCs get a turn whenever it is appropriate. Is this realistic? No. But the PCs are the heroes of the story; they need to be heard; this is how they get heard.
These last two points might feel a bit artificial but you don't have to dwell on them or point them out... just act on them. No player has ever come to me after a session and said, "You know, that was a great session, except for when we talked that mob into not killing the sheriff's son. I really feel like we all should have had less input into that. And next session when we report to the town council, they should totally run right over us and not let us speak."
Not every line of dialogue needs to be spoken. Once you've established the parameters and the personalities, it's okay to summarize several minutes of dialogue as something like, "The Mayor and the Senior Alderman bicker back and forth for a few more minutes. It looks like a little over half the council sides with the Alderman, when one of the Mayor's cronies asks you all a pointed question...." (This is probably a little too generic, but you can summarize the details of what they're arguing about, too.)
The point is not to ration out spotlight in actual lines of dialogue, but in amount of time focusing on PCs vs NPCs. You are will within your rights to summarize your NPCs but to ask for more specificity and direct dialogue (if that's what you want) from your players and PCs.