If people choose to have an animal put down it is illegal for a vet to accept the animal and then not do as instructed by a client. They can refuse to put the dog down and thus not enter into a contract with the client, but given the alternatives, a vet is a very humane way to do this, for whatever reason. It's much better than leaving the dog at the pound if it is unlikely to be rehomed, only to have it put down with large batches of other dogs. r to dump the dog in the countryside, where it may be hit by a car, attacked by other animals, or starve to death.
My vets would generally try to address the issue at hand or steer the person towards a rescue if possible or try and help them rehome. In the case of cavaliers, I have several vets who will ring me if any are brought in to be rehomed or as has happened in a few cases, to be put down. The vets have the owner sign the ownership of the dogs over to them however so that they have the legal right to them.
Unfortunately there is always a massive surplus of animals people decide they don't want and which no one else wants either or which can;t be rehomed in time, and that surplus gets put down. Labs, as lovely as they are, are one of the drastically overproduced dogs (everyone wants the cute puppy but not the large strong adult) and they are very hard to home in many places (in Ireland they are extremely hard to home along with greys, lurchers, collies and staffies). So few rescues can take them in or are willing to take them in. But people continue to buy puppies they then grow tired of when they become adults, or fail to spay and neuter their own dogs who are allowed to roam, or get pregnant and are allowed 'just the one pregnancy', adding yet more puppies into this cycle.
There is a surplus of cavaliers too -- no one needs 'just one litter' and if a dog isn't neutered it has every potential, male or female, to add to the problem -- hence Dog's Trust's strong campaign on this issue in the UK and Ireland. I am only just able to manage the numbers that come in to me (which incidentally is a larger number annually than several of the US breed rescues combined :yikes). If more people started contacting me I would have to turn dogs away. And if I had anything remotely close to the numbers of labs handed in to pounds and vets I couldn't possibly cope.