I have had a similar issue, but I'm not sure if it has the same cause as yours. I am using the Steam version (on Windows 10 with a Ryzen 5 2600 and GTX 1660 Super) and I was able to get the game to load by opening the launcher, logging in, exiting Steam, and then hitting Play. Steam shows a pop up that says it is waiting for FFXIV to close, but eventually closes even though FFXIV launcher is still running. The FFXIV launcher shows a pop up asking if you want to quit, so you can select "no" and still hit the play button.
I was able to reproduce this result reliably: if Steam is running, the game won't load. If Steam is not running, the game loads. It seems clear (in my case, at least) that there is some sort of interference between the game and Steam that is preventing the game from loading. I have updated drivers, the OS, reinstalled the game, all the usual things. I even tried reinstalling Steam. The only things I have not tried are reinstalling Windows and checking the RAM for errors. I have had some intermittent problems with this system, so it's not impossible that there is a hardware issue, but the existence of the workaround leads me to believe that there is a software issue with Steam or FFXIV or both. After a long chain of emails with support, they recommended I make a post in the bug report forum.
Anyway, I'm posting this information here in case it helps someone else. I've seen some recommendations to kill Steam from Task Manager, but that did not work for me because Steam just starts itself back up again. I was surprised that I could exit Steam while the launcher was running because I thought Steam would forcibly kill any games that were running if you tried to close it, but apparently that is not the case.
Also, as regards DirectX (again, in case this is helpful to anyone)...as far as I have been able to determine, the current version of DirectX is 12 and it's included in Windows updates. There isn't any way to roll back to a previous version. There is an end user runtime engine you can install directly, the purpose of which is apparently to provide some libraries that might be needed by older games that use older versions of DirectX. However, Microsoft claims that DirectX 12 is backwards-compatible with previous versions (which is a very on-brand for them).
My understanding of the way this is supposed to work is that the game communicates to the operating system which version of DirectX it wants to use. For games that are launched directly from Steam, there are ways to force it to use certain versions of DirectX: some games will have a popup that asks you how you want to launch it; you can also manually change the launch options in Steam. But I don't think this does anything for FFXIV since it has its own launcher. Since my game will not start when Steam is running, but will continue to run if I restart Steam *after* I've loaded the game, my speculation (assuming the crash is DirectX-related) is that Steam is somehow causing FFXIV to fail to initialize DirectX correctly. I haven't seen any DirectX-related errors in the event log, though, and dxdiag comes back clean, so it's not super clear that DirectX is even the issue.
Anyway, not sure what platform you are on, but here's hoping the patch fixes things for you. o7