I had the same issue and tried logging in dozens of times over 40 minutes while looking for solutions. It seems to be an issue with steam specifically, and the way to fix it is by:
-Logging in but not hitting the play button
-Shutting down steam via task manager
-Pressing play should launch the game like it did for me.

Found this solution from a reddit user named u/malavalon (can just google "ffxiv malavalon black screen" to find the thread).