Original Answer: I'd imagine there are a lot more things that need to be done to complete a modern movie. In the 1920's you didn't have to deal with lawyers to fight for rights to use different intellectual properties. Deal with executives who want to change the view of a director. Deal with expensive and tough to work with actors. Not to mention the amount of editing and refining that are done in movies now. It would also help that movies from the 1920's where silent, and didn't often last for longer then an hour. High budget films that are 2-3 hours with 500+ people working on it take time.
After looking further, while there appears to be this perception that older movies where quicker to produce this may not be the case. See Charlie Chaplin and "The Gold Rush", which was an
elaborate production, costing almost $1 million, included location shooting in the Truckee mountains with 600 extras, extravagant sets, and special effects. The last scene was not shot until May 1925, after 15 months of filming.
Meanwhile a more modern movie like Interstellar started filming on
August 6, 2013, in the province of Alberta.
and
Filming concluded in December 2013
meaning it only took about 4-5 months to shoot. So with this evidence it might be a misconception that older movies were quicker to produce. It might come down to the overall scope of the film, as well as the difficulties within the film's production.