Hi and welcome to the forums, jlcschulz!
i'll take a stab at the 'early entrance' subject...
I don't even see early entry as an option to buy directly from the
Vatican website. The information I found so far says only private tour
companies offer early entry. Is this correct?
You are correct: the Vatican itself doesn't offer any tours that enter before the general public.
To my knowledge, aside from a couple of quite expensive options there are really no true 'early entrance' tours anymore like there were in years past. Those really did get their tours into the museums before the doors opened to the general public. When visiting hours, entry fees and tours on offer changed this past January, those previous 'early' tours pretty much went away. Yes, there are independent companies advertising them but when you get into details, it becomes apparent that 'early' just means 'among first in' at the 8:00 hour when doors open to everyone.
Here's an example of one from the Tiqets website:
https://tinyurl.com/7y87w5x9
As you explore the details, you'll see that the 7:30 meeting point at Piazza della Città Leonina is some distance away from the entry to the museums (on Viale Vaticano). So, by the time the group gathers, walks to the entrance and passes through security (everyone goes through the security checks) they won't be getting in any earlier than other groups or the general public; 'early' just means the 8:00 hour when the doors open. That doesn't mean that there can't be benefits to being amongst the earlier groups, such as mornings being cooler in the summer versus than later in the day, but just be sure to read ALL of the text closely if choosing a tour, OK? It's good to know what to expect. :O)
The couple I mentioned earlier that really do get in earlier are the Key Master's and 'Alone in the Vatican" tours; titles can vary. They are offered by independent companies, for very limited dates, and run $ 700 - $1,600 or so for 2.
Tours booked through the Vatican's site do not access St Peter's. If you're looking at tours through independent companies, you'd have to ask them about accessing the basilica on Wednesdays when the Pope is out of town. The Walks of Italy tours that include the basilica state that the interior passage is "subject to other unexpected closures" in addition to Wednesdays. While I think that's probably rare, it can happen so nothing is 100% assured!
Sorry this was long but there's a lot of outdated info on the net and in the forums regarding tours that are no longer available - or not in their previous forms, anyway - so hope this helps you make your choice! Oh, and most of us advise booking directly with companies who actually conduct their tours versus through 3rd party consolidators like Viator, Get Your Guide or Tiqets.