Is there more information about this?
This is correct to the best of my knowledge; the actual opening of said mail should have been handled by the White House Office of Presidential Correspondence, established in 1897.
https://en.wikipedia.org/wiki/White_House_Office_of_Presiden...
I was a bit curious about what this process looks like in relatively modern times, and found this video from the President Obama White House YouTube channel from 2009:
Inside the White House: Letters to the President
https://www.amazon.com/Eisenhower-in-War-and-Peace-audiobook...