From the client standpoint it seems obvious, I want to do this to save my pet, all you have to do is let me pay it over x number of months or weeks. And since there is nothing physically or legally prohibiting the vet clinic from making that happen, then that's how it should be.
Now, look at it from the veterinary clinic's point of view. If you don't have a savings account, a credit card, friend or family member who can loan you the money, why would the vet clinic? Why is it their responsibility to finance the medical emergency you were fully unprepared for? What if the emergency stems from client ignorance or irresponsibility? What if it's an unvaccinated puppy with parvo, an intact female with pyometra, a free roaming dog hit by a car? The veterinarian should still give you special treatment? And what if it's a pet that has never received any veterinary care in its entire life, but now, suddenly, when it's an emergency that pet means "everything" and is a beloved family member? And what about when it's a dog that is just going to be put back out in the yard on a chain once it has been treated? Now the veterinary clinic gets to determine who is "deserving" of payment arrangements? I'm sure that will go over like a PR nightmare.
And who decides the terms of the payments? The client? What if they want to pay $10/month on their $1000 bill? Under any other circumstance you need to qualify for financing, and someone else tells you what the terms, and in most cases, the interest, will be. If a vet clinic needs to employ a financial department then guess what, costs are going to go up even more.
And I get that some illnesses are unexpected, not preventable, and expensive. But if you choose to get a pet then you also need to think about what you will do if such a situation arises. You can get pet insurance, you can set up your own emergency fund and save money every month, apply for a credit card or Care Credit (put it away in a safe or freeze it in a block of ice if you have to but at least it's there if you need it). It's your responsibility, just like when the car breaks down, the house needs a new roof, etc.
Like it or not we all make pet owning decisions based on money. It might be how many pets we own, how well they are cared for or even whether they live or die. At some point we all need to decide how much money we would be willing or able to spend to extend or save a pet's life and where that money will come from. That amount is going to vary from person to person, pet to pet, and depend on age, prognosis, and quality of life. But it is something you need to think about in advance.
And I'm not saying a person needs unlimited funds or a minimum amount of money saved before getting a pet. I've seen a lot of clients on a tight budget do a great job of caring for their pets. It might mean spreading out routine expenses and saving or budgeting for larger procedures like dental cleanings or lump removals. And often, if these clients have a good relationship with their vet the clinic will work with them if an emergency arises. But a person also needs to be prepared to make difficult decisions if the need arises. Having a pet is a privilege, and potentially curable pets get euthanized all the time for financial reasons. Veterinarians don't like this any more than you do, but that is the way the world works.
And it doesn't make you a bad pet owner, sometimes you just have to prioritize. We all do. If something hugely expensive happened to the barn cats that I couldn't fix myself I would put them to sleep. If Squirt, at 18 years old or more, needed $7000 colic surgery I wouldn't do it. These pets are loved, have a good life and are taken care of. But I have to draw the line somewhere. It would be hard and I would be sad but I also wouldn't expect the referral center to let me make payments of $100/month for years on end because that would make it convenient or feasible for me to afford something outside my comfort level. On the flipside, when Icy got sick I handed them my credit card and dealt with the payments later. It wasn't easy or convenient, but it was our decision to treat her and that's why we have the credit card. I didn't have the money up front, but I had a way to pay so she could get the care she needed.
A person should also, ideally, think about the type of pet they are getting, the problems they are predisposed to and how they would deal with them should they arrive. Granted, you cannot predict everything, but running blindly into pet ownership with the idea that the veterinarian will save your pet at a moment's notice regardless of your ability to pay is foolish and irresponsible.
I know some people feel strongly about this issue, and hopefully this gives you a perspective on why things are the way they are and a little on how to be prepared for an emergency-both financially and emotionally.