Was this a requirement or a driver preference
Stock Car Racing History
That pic is from the inaugural World 600 in 1960. The newly christened Charlotte Motor Speedway built by Curtis Turner and Bruton Smith was hardly ready for prime time. The pavement was chunking badly. All 3 Petty cars ran the screens to help protect their cars as you say - and their heads!
Here is another pic of Lee racing alongside teammate Bobby Johns.
And here is Richard racing alongside Jack Smith.
As it turns out, the Petty Plan was for naught. Both Lee and Richard were DQd by NASCAR along with Bob Welborn, Junior Johnson, Paul Lewis, and Lennie Page for accessing pit road inappropriately or something like that. Bobby Johns was not DQd and salvaged a 3rd place finish for the team.