That's great, except that most shelters and rescues are privately run and receive NO funding from the state. Absolutely none. The ones that do receive funding are often under-funded because, in the (paraphrased) words of a government official in one city: "We have murders to worry about here."
If you'd like to offer some real, creative solutions instead of wishing for state sponsorship that'll never come, I'm sure the shelter community will be happy to listen.
On a more serious note, though: more money is only helpful to organizations that can use it effectively. Unfortunately, many organizations are just not as well run as they could be. Also, veterinarians are not usually the ones in charge at the shelter. I'm sure there are exceptions to this, but normally they're working under the rules and mandates set by the board of directors or operations director - just like everyone else. They may or may not be making decisions on who stays or who goes.
Double unfortunately, many staff members at shelters are not well trained where euthanasia is concerned. In my opinion, it's better for a qualified person (like a veterinarian or well trained tech) to be euthanizing animals than someone who was barely trained and doesn't really understand what they're doing. I think that euthanasia can be a gift for animals that are suffering - whether that be from a physical disease or from the stress of being stuck in a cage in a shelter with little enrichment for the past 10 months and no prospect of adoption.
I'd like it, too, if we didn't euthanize any healthy animals, but remember that it goes much farther than "a little overcrowding." Animals in overcrowded conditions without enough enrichment or attention develop health and behavioral issues that are really difficult to break. Certainly more than the average pet owner wants to deal with.
Obviously, something needs to be done. I think increases in foster care programs and community outreach are the answer. That and better education for shelter staff on the physical and psychological needs of animals. I also think that vets in these situations are dealing with way too much for me to condemn them as "unethical." It's just not that simple.