Same settings as fuji29832. After using my X-T3 as a webcam for > 1hour, I had the same symptom : "Fujifilm X Webcam" white on black logo. Changing the USB_C cable did not solve the problem. As it seemed to me that the X-T3 was a bit hot, I switched it off to cool it down. I tried again to use it as a Webcam approximately 2 hours later and... everything was working fine again !
For some reason Fuji cameras are quite susceptible to read/write errors on SD cards. I've had multiple issues (incl. lost images) with several different Fuji cameras, so I use the following 'workflow' to reduce the chance of error:
always use Fuji recommended SD cards
use two exact same cards if the camera has two slots
insert them gently straight in and out
format both cards in the camera before a shoot
regularly copy (not move!) all images on the card to your computer
then (re)format both cards in your camera (never on your computer)
do not take out one card to view images on a computer/tablet and then put it back in the camera for further shooting (I share the images via my phone to a Dropbox to view them on bigger screens. Or in the studio I work tethered.)
That way you don't have to select and delete images one by one or by group in your camera, which is a tedious process.
In general: SD cards are not meant for long-term storage of images. Just for shooting and transport to your safe storage on a computer and back-up drives.