So, I love it when I hear that vet clinics are doing much of the vetting/rehoming for strays turned into them as they are saving a space in a shelter. Most shelters euthanize for space. Many animals do not thrive at shelters and could get euthanized for medical/behavioral reasons. I know because I work at one (but not a typical shelter, thank goodness). The vet clinics simply ought to notify the organization to let them know what they took in and when in order to give that animal it's rightful 'stray hold' and chance to be claimed. But if they can keep the animal out of the shelter, they are doing that animal and another animal (that will occupy its kennel) a favor...
It's not legal in all places to hold a stray animal. In someplaces, legally, a stray MUST be turned in to AC (animal control) and/or it is illegal to adopt out an animal that wasn't processed through AC (in other words, not all ordinances are set up to enable off shelter housing with notification to AC.) This is a legal issue that I have had the challenge of dealing with as a county employee, a non-profit employee, a humane society volunteer, and a vet clinic employee. It is very important to know the legalities of your state, county/parish, and local government. Violating these ordinances can be very problematic for a business owner.
Another consideration here is that by taking stray animals into a vet clinic, you may be exposing other animals to things like respiratory infections, parvo, etc. If you have an isolation ward, you may be utilizing an important resource that you might need for patients. This is costly on a number of levels. That isn't including potential treatment of the animal.
Follow all of that by the cost to care for these animals and the conditions they may live in. Someone needs to clean, feed, and exercise these animals. Especially in clinics that don't board, that can mean an extra person on off days (such as Sunday), not to mention extra staff hours every week. These are real costs that can be damaging. Also, many clinics are not set up for the appropriate residential care of animals (I realize many shelters aren't either) but if the animals' entire existance is going to be limited to a small cage, with limited interaction with people and limited exercise...well, for some animals that can create temperment issues.
Let me share a recent experience with this. Someone dumped 4 kittens on our doorstep. Our vet took them in which was fine and we started vax. Within a week, we found ringworm on the kittens. However, the kittens had already had 'exercise' time out in the wards and in the exam rooms (places that could be closed off.) Ringworm is contagious; within 3 weeks we had over a dozen client's pets develop ringworm, about half of which were inside only animals without other exposure. I am glad our vet was honest and treated these cases for free, but it was costly in terms of treatment, time, energy, effort, and customer relations. It is hard to explain to an upset parent that thier inside only cat probably got ringworm at our clinic, and a week later, get the call that thier child has it as well.
In the end, it took 3 months to place 3 out of 4 kittens. We were never able to place the last one, which did go to AC. Realisticly, we weren't set up to quarentine strays, to provide adequate exercise, and/or to display kittens for adoption. It was a very costly experience which damaged our reputation with clients, right before a pretty severe economic downturn in the primary industry that supports this region (banking.)
If a vet has the ability and set up, is within the legalities of thier locale, and wants to provide for strays, then by all means they should. Realisticly, in high kill areas, this can mean a vet who is inundated with animals (or now, in areas where people can no longer afford thier pets.) I would suggest anyone doing this routinely to set up a legal non-profit. I don't think there should be an expectation that vets become a center for animal sheltering, or should feel bad abou utilizing shelters for strays and abandoned animals.