how to most efficiently and best visit the Vatican/Sistine Chapel/St. Peter's Basilica

We did a tour called the Early Entry Express Sistine Tour. It was fabulous. We caught a taxi to the Vatican from our hotel as it was still dark. We met our tour group at around 7:30am from memory. That was the only worrying part as there were lots of tour groups, but we did find their sign. Make sure you allow enough time.

We were in before 8am, and the guide literally raced us through the Vatican Museums, but taking the time to point out the most interesting attractions for us to come back to.

We got 20 minutes in the Sistine Chapel with just our group of 40 (I think). It was simply incredible. Once our time was up, the guide showed us the "secret" door that gets you into St. Peter's without having to go back to the beginning.

We then went back into the Vatican Museums and had as much time as we wanted. We probably spent two hours or so in there, before returning to the Sistine Chapel. My goodness what a difference! It was literally mobbed with wall to wall people.

I am so glad that we had that 20 minute whereby we could lie on the floor and look at the ceiling and spend time just enjoying the space.

We pushed our way through to the "secret" door, went through that, and we were at St. Peter's.

It was a perfect way to see it all and I highly recommend it.

Once our time was up, the guide showed us the "secret" door that gets
you into St. Peter's without having to go back to the beginning.

Not sure how long ago you did this but that door has been unavailable to independent tourists for some time, and not even every tour group is now allowed to use it. While you used an express tour of some sort, you were essentially sightseeing the museums on your own after you left the Sistine so that door is likely not available to tourists now doing the same.

A lot changed at the Vatican after they did away with practically all early-entry (e.g. admittance before general entry time of 8:00 AM) and some other types of tours in January of this year. There are a few tours that still get into the museums before 8:00, like the "Key Master" or "Alone in the Vatican" tours but their price points, at hundreds of dollars per person, are more than most visitors can afford.