It turns out that director Joss Whedon had an idea for a Batgirl movie after all.
Whedon previously claimed back in February that he just didn't have a story to tell, but he revealed at the Avengers: Infinity War premiere that this really wasn't the case. He also explained to Variety why the project fell apart after being in development for over a year.
According to the director, Whedon had envisioned his Batgirl movie to fit into a very specific point in DC's film universe. Unfortunately, Warner Bros. didn't want to wait around for this movie to come to fruition, so the DCEU went full steam ahead. Given the fact that a Batgirl film is not considered to be priority by the studio, this should not be surprising.
Whedon saw that his idea for the film no longer fit into the universe, likely because it was supposed to take place before something like Justice League happened. Instead of attempting to alter the project to fit the state of the franchise, Whedon chose to drop the project and move onto other things.
DC Films hasn't given up on the potential for a Batgirl film, with Christina Hodson hired to write the script. Hodson previously wrote the Transformers Bumblebee spinoff film that is expected to be released on December 21, 2018.