Sorry I couldn't at least direct you in a direction towards a fix for your game issue but hopefully someone at Bethesda will be able to figure out what is wrong or at Bethesda Forums for DOOM.
Take care and good luck.
I had the same issue with my RX 6900 XT recently, I forgot to reply how I fixed it.
Mine wasn't showing Vulkan supported in GPU Z like in the above picture, though.
It was showing up to date in Adrenalin and wouldn't update.
I had recently done a DDU clean when I installed the card at the end of January. Vulkan worked fine in RDR2 but I guess that worked differently.
On closer inspection my Adrenalin wasn't the latest version it was from Dec, there was a new version listed online in Feb.
I may of broken something the first time I did a DDU clean. This time I uninstalled the Radeon software via Windows and reinstalled and it was working again.
Anyway it may not be the solution to your problem but if you recently installed your drivers something may of gone wrong so worth trying a second time.