I finally discovered a fix for the black screen bug when launching Titanfall on PC. I tried multiple things to no avail, but what finally worked for me was this:
Go into your Titanfall directory. Select all files in this directory (exclude folders) and delete them.
Launch the game. The missing files will be downloaded from Origin. Once this completes and the game launches, be patient. You will see a black screen for 10-15 seconds and then you'll see the familiar Respawn logo.