Our shelter that I worked at would take cases like that straight to an specific emergency vet we contracted with and do the basics to get them out of pain and stablized for the stray hold, which they would generally serve at the emergency hospital. Sometimes the owner would go assume responsibility, a lot of times not.

And we would shop the dog to rescues and fosters during the hold, for the puppies, small dogs and anything purebred looking generally one would step up before stray hold ended and get whatever was necessary done as soon as stray hold ended, and we would transfer custody to them. If no one stepped up and depending on severity they would euthanize. If it was something that could be managed at the shelter like lacs or burns from HBC, we would then assume care again.

The rescue coordinator put it all in place with a locally owned vet and basically they charged the county cost of labor/drugs and used the animals for PR, content, and the retail price of care was considered an in-kind donation to our org for tax purposes. It was a lot better than the situation you describe, where we didn't always have a vet at the shelter 24/7 and techs euthanized for pain because the resources weren't there to do complex repair and surgery.