From my experience, certain "troublesome" liveries cause this issue. I've never had this issue with a car with simple or no liveries.
The cars show up in the garage, races, and replays just fine, but for some reason accessing GT Auto seems to make the game ping the servers for the car's livery every time, and sometimes it just won't feel like delivering. The issue often resolves itself after I step away from the game for a while.
Attempting to do anything car related in GT Auto in that state, such as oil or wheel changes, just causes a softlock with an infinitely spinning GT icon. Attempting to load the livery or creating from existing livery will also softlock the game.
I'd like to think I'm on a solid, stable wired internet connection and it still happens occasionally. I'm on the base PS5.
Sadly, as has been discussed in the Livery Editor subforum, there seems to be no known workaround for this to this day; not unless you're open to reverting your car livery back to the original just to do maintenance on it.
If you don't play in online races, you can always hit Retry → Car Settings. Simply bringing up the quick menu to change tyres will reset all mileage accrued in that session along with the associated wear and tear that comes with mileage.
I know none of what I've written is ideal, but I hope this helps nonetheless.