I don't think I understand the issue here. It sounds like on both devices you get a home screen shortcut added, although the latter is pointing to the incorrect URL.
The issue with trying to reach https://cohost.org/static is due to a cache issue with an initial release of this update. Removing the shortcut, clearing you browser's cache, and re-adding the shortcut should fix this issue.
Please let me know if there's something I'm missing. We don't have devices to repro or test this specific case (we have exactly one (1) android device we can use for testing and I do not have the time to switch between custom ROMs to test shit on it) so I can only work off information you or someone else provides.
You're right, it must have been a cache issue. By coincidence, the device without GMS kept adding the same broken cached version, and I fixed it by clearing the cache and re-adding the shortcut.
For whatever reason when I post on Cohost Support my newlines are getting removed, so I'm sorry if this turns into one unbroken wall of text again.
I have two Pixels both running GrapheneOS, one with GMS installed (Google Services Framework, Play Services, Play Store) and one without. The thing about GrapheneOS is that you can install GMS as regular apps and everything they would normally need privileged access to do is sandboxed. The majority of features work. Notably for this bug report, WebAPK is broken, which is the feature Play Services will use when you want to add a PWA to your home screen and you get the option to "Install app" instead. (It's basically Electron.)
As of the update that just went out within a day or two that fixes the missing icon on Firefox, I now get the option to "Install app" instead of "Add to home screen" when I open Cohost in Vanadium.
On the device with GMS, I can add the Cohost PWA to my home screen, and it won't install a WebAPK but it will still add the regular PWA to my home screen.
On the device WITHOUT GMS, the task fails successfully and I get a PWA that HTTP 404s trying to reach "https://cohost.org/static".
I really don't want to install Play Services just to restore the PWA (I was an arrogant fool and deleted my old home screen shortcut first without confirming the new one would work). I know I've used other websites that support being installed as a WebAPK but fell back to a regular PWA just fine, so I'm assuming this is a bug on Cohost's side. There might be a way to check if the website is running in a WebAPK before switching to the static version of the assets.