Things like The Avengers movies and Star Wars movies are notoriously secretive. Scripts are monitored, and delivered on tablets, with watermarks. Then promptly deleted after a certain amount of time. Or actors are brought into a guarded room with no technology, to read a script and then has to return it before they leave.
Methods were used for Avengers: Infinity War and Avengers: Endgame. Most actors only received the pieces of the script they were in. In fact, the actors weren’t told the ending of Infinity War, until day of. They were in a van on the way to set and they were told what would happen.
Endgame was probably even more severe, and as more and more people see the movie, the Russos are more open to talking about it. In a Q&A at the Smithsonian in Washington DC (and reported about on Reddit), the Russos said only Robert Downey Jr. read the entire script before filming. The reasoning for this makes sense. The other actors get their relevant parts, they have less of a chance of slipping up in interviews. They used the example that Chris Hemsworth doesn’t need to know what Captain America is doing during Infinity War. Downey getting the whole thing is important for two reasons. One, the entire story of Endgame, even moments not involving him are important to Tony Stark’s journey. The second is that Downey has a history of changing the script and adding moments for the better.
There is a wonderful thread from @cxptainflint on Twitter explaining Downey’s contributions to the MCU. Some of the biggest moments in the franchise were his idea. For instance the “I am Iron Man” quote to close the first Iron Man movie was a Downey invention. So was the “I love you 3000” response Morgan Stark gives Tony in Endgame. It is a frequent exchange he has with his children. Even the more subtle stuff, like using the pen set has an olive branch to Steve in Captain America: Civil War was Downey’s idea. He is known – and celebrated – for it. Feige loves it, and so do the writers, who get credit for it.
So with that reputation, it makes sense that Downey gets the entire script. He would be able to improvise (and improve) things better with a full picture in his brilliant head.
As the Godfather of the MCU, it makes sense that Downey would have that honor. Besides, for all intents and purposes, this is the last MCU script he will ever read (though I do feel he can cameo as an A:I interface/voice like JARVIS if Marvel ever decides to do a Riri Williams Iron Heart flick, which they absolutely should). So it is a nice gesture for them to close his story on.