Movies were in 4:3 until television came along. When TV came along and got popular in the 50s, hollywood wanted to differentiate themselves from television and thus invented the completely arbitrary 16:9 format. Sadly, this and even more extreme unnatural formats stuck around and became so popular even TV is adopting them now.
It was never about directors, it was about being "different from TV"
The spirit of abysmal despair

I've always thought that the reason was the aspect ratio of the frame. Until the 1950s and 1960s, the images in movies were in the 4:3 ratio. (Just watch an oldie from the 1940s like
Casablanca
or
The Big Sleep
and you'll see that the images were almost a square, like a TV screen.)
I thought that the advent of widescreen allowed directors to stage wider images and to put an action and a reaction (i.e., a gunshot and a falling body) into the frame at the same time.
Eastwood may
say
something like,
"Sergio Leone was the first to "
but since when is he cited as an oracle on developments in film?

Ive read that remark by Clint Eastwood too. It must have been a rule they followed on
Rawhide,
but it was never a rule enforced by the Production Code. In addition to the movies you list, the following movies include scenes in which the shooter and the person shot are in the same frame:
The Strange Love of Molly Louvain
(1932). This is admittedly Pre-Code.
To Have and Have Not
(1944). But only the leg of the person who is shot is in the frame.
Between Midnight and Dawn
(1950). There are two different scenes in which this happens.
Shane
(1953). Two men are shot in that fashion.
The Manchurian Candidate
(1962). At least two people are shot this way, maybe three, if you include a scene in which an arm partially blocks the image of the person being shot.

If you watch most American western gunfights in films made before the late 1960s, it is indeed correct that the director will cut away from one man drawing a gun to show the other man firing. However, there is no law or anything in the Hays Code that requires this. It is just tradition and, as correctly pointed out above, there are numerous examples of this tradition not being followed at all. Even before 1964.
So, basically, the whole idea that Leone invented something new here is nonsense.

Rio Bravo (1959)
The Comancheros (1961)
The Man Who Shot Liberty Valance (1962)
I don't recall a moment like this happening in The Searchers (1956) so the code must've been still in full swing at that time.But somewhere between that and Rio Bravo it's changed.

Fill your hand you son of a bitch!

It was the result of something called the Hayes (or Hays) Code, whereby the weapon and victim could not appear in the same fram together. Leone knew nothing about this so he did what he wanted.
I do not know when the Hayes Code began nor ended. Which of John Waynes films did you see this happening in? It would be interesting to know their release years.
Never defend crap with "It's just a movie"
http://www.youtube.com/user/BigGreenProds

It may have been a TV thing