This is hilariously untrue. He was very well known for his “a time for choosing“ speech in 1964 introducing Barry Goldwater. It had the same impact as Obama‘s speech before the Democrats in 2004. As a union leader he gave many, many well received speeches that he wrote himself. He was politically active as a union Democrat for decades before becoming governor of California and wrote scores of speeches at that time.
From about the 1940s to the 1960s he made many radio addresses that he wrote himself. He was politically very active, touring the country and doing a lot of speeches on his own.
As a president he absolutely had the best speech writers of his time, but he went over each speech meticulously and gave feedback the writers claimed was expert and welcome.
One of his speech writers actually published a book that showed photographs of Reagan’s own handwritten notes for his speeches. There are thousands of them in his presidential library.