“Don’t do anything yourself that someone else can do for you” is a quote that was attributed to Calvin Coolidge (1872-1933) in June 1929. In The American Magazine and in his Autobiography (both 1929), Coolidge was quoted as saying:
 
“In the discharge of the duties of the office there is one rule of action more important than all others. It consists in never doing anything that someone else can do for you. Like many other good rules, it is proven by its exceptions. But it indicates a course that should be very strictly followed in order to prevent being so entirely to trifling details that there will be little opportunity to give the necessary consideration to policies of larger importance.”
 
The Coolidge presidency (and his quotation) has been popular with conservatives such as Ronald Reagan.
 
   
Wikipedia: Calvin Coolidge
John Calvin Coolidge, Jr. (July 4, 1872 – January 5, 1933) was the 30th President of the United States (1923–1929). A Republican lawyer from Vermont, Coolidge worked his way up the ladder of Massachusetts state politics, eventually becoming governor of that state. His conduct during the Boston Police Strike of 1919 thrust him into the national spotlight and gave him a reputation as a man of decisive action. Soon after, he was elected as the 29th Vice President in 1920 and succeeded to the Presidency upon the sudden death of Warren G. Harding in 1923. Elected in his own right in 1924, he gained a reputation as a small-government conservative, and also as a man who said very little.
 
19 October 1906, The Plain Dealer (Cleveland, OH), pg. 12, col. 5 classified ad:
Do nothing yourself that someone else can do for you just as well and at less cost.
INTERSTATE BUSINESS EXCHANGE,
318 Citizens bldg.
 
Google News Archive
19 June 1929, Newburgh (NY) News, “Coolidge Was Hard Worker as President,” pg. 1, col. 3:
New York, June 19. (UP)—Calvin Coolidge lays down a group of guiding principles for men who are at the head of great organizations, in an article to be published tomorrow. They include:
 
“Don’t do anything yourself that someone else can do for you.”
 
Google Books
The American Magazine
Volume 108
1929
Pg. 15:
In the discharge of the duties of the office there is one rule of action more important than all others. It consists in never doing anything that someone else can do for you. Like many other good rules, it is proven by its exceptions. But it indicates a course that should be very strictly followed in order to prevent being so entirely to trifling details that there will be little opportunity to give the necessary consideration to policies of larger importance.
   
Google Books
Autobiography of Calvin Coolidge
By Calvin Coolidge
New York, NY: Cosmopolitan Book Corp.
1929
Pg. 196:
In the discharge of the duties of the office there is one rule of action more important than all others. It consists in never doing anything that someone else can do for you. Like many other good rules, it is proven by its exceptions. But it indicates a course that should be very strictly followed in order to prevent being so entirely to trifling details that there will be little opportunity to give the necessary consideration to policies of larger importance.
       
25 July 1931, Hamilton (OH) Daily News, “Coolidge Still Keeps ‘Em Guessing” by William C. Murphy, Jr., pg. 10, col. 5:
Also, he has composed some maxims for the successful conduct of the presidential office. There is one of which he says “It is more important than all others. It consists in never doing anything that someone else can do for you.”
 
But he has added that, “while it is wise for the president to get all the competent advice possible, final judgments are necessarily his own.”
 
Google Books
Executive Thinking and Action
By Fred DeArmond
Chicago, IL: L.R. Wolfe
1952
Pg. 21:
“If you want a thing done right, do it yourself,” is a venerable copybook maxim, but a better one for the executive is: “Never do anything yourself that someone else can do for you, even passably well.”
 
Boston (MA) Globe
The art of the positive
Ronald Reagan rewrote the definition of politics with a relentlessly upbeat approach
December 18, 2005
President Reagan: The Triumph of Imagination
By Richard Reeves
Simon & Schuster, 571 pp., $30
(...)
Ronald Reagan came to Washington as the most thoroughly conservative president since Calvin Coolidge nearly six decades earlier. Indeed, Reagan embraced Coolidge as his official role model, hanging his portrait in the Cabinet Room and rereading his autobiography. Reeves says that Reagan, who loved to delegate, likely enjoyed one particular paragraph from Coolidge: ‘‘In the discharge of the duties of the office there is one rule of action more important than all others. It consists in never doing anything that someone else can do for you.”