Back when people met up with large groups of friends and/or strangers in person, I was one of the default rules teachers at my big monthly game group for any game I knew because I had a reputation there for being good at it.
I usually just try to do as much big picture as possible then slowly focus it in. For the most complex games (I play a lot of heavy euros and the like), this sometimes would mean multiple levels of focusing in. A setup similar to others have described for thematic world building and end game goal “Here’s who we are and what we’re generally doing in this world and what we’re trying to accomplish to win.” Then there might be an introduction to the various systems of the game and how they fit together “Over here is a ton of different ways to gain resources. I’ll come back to all the different action spaces and explain the icons in detail in a minute, but this is generally where you’ll go to get stuff and how you’ll do it. Now this over here is where you go to hand in basic resources to create more advanced resources, again, we’ll come back to the details later. And over here is where you spend those advanced resources on upgrades for your individual player boards” or you know whatever for the game in question. Once all the different systems and how they interact with each other are in place, then I go back over each one with a quick reminder of how it fits into the big picture and give the more details on what to do there and the iconography and whatever else is necessary to know about it, pointing out any player aides and board reminders available that pertain to it specifically.
My particular style is highly prone to the interrupter with “What about this rule? You haven’t mentioned this tiny detail yet!” Well, no, I haven’t, because the new people won’t understand that tiny detail yet because they don’t know the three high level things that detail interacts with yet. I’ll get there! Because of that I would tell others who know the rules that I find it is best to have one person explaining the rules. If that is me, if I get something WRONG, interrupt me and correct me. Absolutely. If you think I forgot something, please don’t interrupt. Just wait. I’ll check in later to ask if I forgot anything (usually at the end of each detailed section because then people get it) so just hold that thought because I will probably get there. That is most often directed at my husband who just can’t help himself, so it’s fine.