The answer is that usually the director had no say over the music score and if you saw a movie in two different theaters you would hear two different music scores. There are exceptions though.
In 1915, director D. W. Griffith hired Joseph Carl Breil to write a score for his epic but racist film Birth of a Nation. Many times, studios would commission a music score for a film that could be used if a theater owner wished to use it. Short films rarely had a music score.
For example, here is a press book for Womanhood, The Glory of a Nation (1917). In the lower corner, it says,
COMPLETE MUSIC SCORE
For use with the presentation of "Womanhood," a synchronized musical score is available, suitable to meet the requirements either of a complete grand orchestra or one pianist.
Even in this case, the director may have had some input on the music but it was compiled by studio composers.
Here is a program from Wings (1927) screening at the Erlanger Theater in Chicago. Paramount commissioned J. S. Zamecnik to write the score. The Erlanger was a large theater with a big orchestra, so it was probably a wonderful experience. And the Paramount BluRay of Wings does have his score recreated on the disk.
There are several reasons why there was no official score to most films. Unless it was a big blockbuster that would play a theater for weeks, most films only showed a few days, or even just one day in a theater. Having to purchase a new score and rehearse it several times a week was not practical.
Only big theaters had big orchestras. Theater organs only came into standard usage in the mid-to-late 1920s. Big theaters would have the organ play afternoon matinees, because it was too expensive to have a full orchestra work all day.
Smaller suburban theaters might just have a small group of musicians, like this one in Geneva, New York with a sextette. Smaller theaters might just have an organist or a single piano player. They might be playing the piano to accompany a film that they had not even seen yet.
Another problem was that silent films had no set projection speed. Theater owners might run them faster than intended just to get in an extra show. If the film runs faster, the "official" music score would not fit.
And finally, silent films were subject to censorship in different areas and some scenes or intertitles might be cut. This would make the music score be longer than the movie.
Photoplay music was available from some composers, like the same J. S. Zamecnik who had moods for chases, love scenes, crime scenes, comedy scenes, etc. If a pianist or group could memorize these music themes, they could score any film. Rodney Sauer of the Mont Alto Motion Picture Orchestra has some information on photoplay music on his website. His group has scored many silent films and plays at film festivals.
Ben Model is another piano accompanist who has scored dozens of silent films on disk. He will tell you that many times at festivals he has had to provide accompaniment even though he had not seen the film. His music blog describes all kinds of silent film music topics.