A year ago when I was in Venice, visitors were not being allowed into St. Mark's Basilica. When you bought your ticket you actually were only allowed into the museum. So the only way to see the basilica itself was to attend mass. Does anyone know if that is still the case?
It’s open for visitors
I wonder if you were there on a Sunday?
On July 26 Bill from Lubbock TX posted an in-person report about his experience in Venice. He said the Basilica itself is no longer free, so the line to get in was very long as each visitor decided which ticket to purchase. I believe there are separate charges for the museum, the Pala d'Oro and the treasury, so you can see that the multitude of possible combinations could lead to time-consuming purchase transactions.
There was more information in Bill's post that could be helpful to you in planning your attack on Piazza San Marco. Here's a link: https://community.ricksteves.com/travel-forum/italy/update-from-venice
The Doge's Palace next door is also a great place to walk through!
What you’re describing are the entry rules for Sunday morning. The rest of the week the basilica is open for regular tourist visits.