A pulpit is high, prominent place from which a preacher or minister delivers sermons. A bully, to our modern ear, is a person who uses their advantage of strength to harass or coerce people who are much weaker. It's therefore natural for a person unfamiliar with this term to imagine that a bully pulpit is a place or position that affords a bully a prominent opportunity to act out and do bullying.
But that's entirely wrong.
The 'bully' here is not the noun that is in the minds of so many people these days, but rather the adjective that once popularly meant 'good' or 'excellent'. ('Bully for you!' or 'It was a bully adventure!') So the true meaning of a bully pulpit is simply some exceptional position or place of authority from which to speak out and be widely heard as authoritive on any idea at all. That's what Theodore Roosevelt meant by it when he said, "I suppose my critics will call that preaching, but I have got such a bully pulpit!" He said that about the U.S Presidency when he was the President, back when the 19th century had just become the 20th century. Teddy was fond of saying 'bully' to mean 'first-rate' or 'great'.
The term was popular for some number of years after the press had got hold of it, but it then pretty much fell out of use until the 1960s, possibly because it was used by Arthur Schlesinger and Theodore Sorensen in books about the Presidency of John F. Kennedy. It caught fire again around 1985, climbing sharply in frequency of use until about 2005, when a small decline began. The expression remains in current use. I have to wonder how many of us misinterpret its sense.
Frequency of use, 1800 to now