This is the same process here. A dealership will assess if it's a warranty issue, then elevate it to Hyundai Assurance. Sometimes you have to go to Hyundai direct to file a complaint against the dealership when they make a wrong assessment.This isn't a knock here but I think we need to clarify as you mention this all the time.
Hyundai Assurance is just a US only thing and not relevant to everyone else outside the US (generally speaking).
Our process in Europe:
We have a Warranty.
The Dealership decides if its a Warranty claim. Some accept everything, some wriggle out of it.
If the dealership says its not a Warranty claim then the owner has to contact Hyundai head office in their country and bitch and moan to them to try and get things done. This takes time and is a massive ball-ache. Some countries head offices are better than others.
Or go to another dealership and try again.
I wish we did have something like Hyundai Assurance but we don't (unless I've been living under a rock and not realised we do)
Basically we have to do all the "Donkey Work" ourselves if dealers are being awkward.
This is why I use the same terminology time and again.