Would anyone like to see this remade with modern actors; such as Kurt Russell or Denzel Washington?
No.

I kept on seeing the late Philip Seymour Hoffman's rather chubby face while watching Welles.
And I reckon Patricia Highsmith saw this movie and thought about the American Harry misbehaving overseas when dreaming up the Tom Ripley stories.
And doesn't Harry Lime refers to Holly as 'old boy'? And doesn't that also occur in the 1999 version of
The Talented Mr. Ripley
.

All of them
Not as famous as the third man, but still classics

I hear ya. But which of those were classics originally? The bar is set higher when the original is already a classic

Scarface (1983)
Ben-Hur (1959)
The Fly (1986)
The Thing (1982)
Nosferatu (1979)
True Grit (2010)
Ocean's Eleven (2001)
The Man Who Knew Too Much (1956)
Invasion of the Body Snatchers (1978)
A Star is Born (1954)
A Fistful of Dollars (1964)
The Magnificent Seven (1960)
Cape Fear (1991)
All great remakes of older films

Off the top. Name a remake of a classic that turned out well?

Why remake a perfect film?