I got a lot of useful insights from this thread (thanks!) and want to share my story as well. I upgraded my iphone X to 13.4.1 on 5/7. Right before the upgrade, I backed up the phone which was with iOS 13.4. The upgrade ended up with the frozen "swipe up to upgrade" white screen. I then started my lengthy and exhausting working sessions with T-Mobile and Apple for the next week or so, tried several different attempts (including restoring to factory and then restoring from my 5/7 backup) and they all ended up in vain. By learning a tip from a reply here that one of you successfully restored from an old version of backup with 13.4.1, I happened to have another backup dated back to January. I tried and succeeded recovering my phone with the Jan backup a week ago. Spent few days backing up the important data especially the images to the cloud photo service, I was discussing with the Apple agent (who is the most recent one of a few that I have worked with on my case and he is very helpful) to retry restoring to the 5/7 backup with 13.4.1 (based on the discussions in this thread, I did not have much hope but wanted to try the luck for a 2nd time). A few days ago, 13.5 was out right before my retry attempt. So, I just backed up my phone, updated the phone to 13.5, restored to factory and then restored it with the 5/7 backup created with 13.4 and it worked. Seems 13.5 resolves the "swipe up to upgrade" white screen issue introduced by 13.4.1.
To me, this is a HORRIBLE software bug introduced in 13.4.1 OS that directly turns non-trivial percentage of phones into "brick". Even worse it happened in the current pandemic time that everyone depends much more heavily than ever on phones for their living.