The short explanation for all of this is that Apple’s own standard practices for firmware updates got in its own way. The Studio Display is, in a loose sense, a really big piece of iPhone 11 hardware. And as such, its firmware access is limited to what Apple makes available for the iPhone 11.
MacWorld theorizes that, after releasing iOS 15.4.1 (which is not currently available for the iPhone 11) on March 30th, 2022, it ceased signing for the now-obsolete version 15.4. In other words, Apple stopped verifying the older version of the firmware, which caused attempts to download it to fail because the hardware could no longer confirm that it was valid, thus the Studio Display refused to install it.
On the plus side, fixing the problem was relatively simple: Apple just had to sign iOS 15.4 again to signal to everyone’s Studio Displays that it was a legitimate download once more. Though it does raise questions over how an oversight like this could happen — with a brand new piece of Apple hardware, no less — and what could be done to keep it from happening again in the future. But at least now we know that if there are problems with firmware installation, we might only have to wait a few days for it to sort itself out.