Convenience Euthanasia request broke me

I am a veterinary receptionist not a technician so I hope this post is allowed.

Earlier today I got a call from an already existing client, and he has 3 elderly cats (about 9/10 years old) that we have seen and treated before. He expressed that these cats were perfectly healthy, but he no longer wanted to care for them anymore and wanted to know if one of our vets could come to his house and euthanize all 3 of them. He said he did not want to take them to the shelter because "no one wants an old cat". I know that kittens and younger cats are more appealing to adopters, but this broke me because I in fact rescued my cat when he was already 6 years old. I became emotional on the phone and went mute for a couple seconds as I was gathering myself and he said "should I call somewhere else because you aren't being very receptive". I apologized and told him I would talk to the vet.

I spoke to the vet and he could tell how upset I was, I absolutely love cats and just the thought of those poor babies losing their lives just because had me in tears. He wanted me to put in a message so that he could call the client after hours to talk about the situation. He is the one vet out of our 4 that always gives the client what they want no matter what. I could tell in his demeanor that he was most likely going to do it for the client.

I am very disappointed that he would even consider that. I know in the state of Tennessee, pets are considered private property, but I thought we absolutely had the right to refuse service? I know we cannot force them to surrender, but I'm sad knowing that we do not even AT LEAST get a chance to try and find a better solution for those babies? Because I know the rest of the staff and I myself definitely would.

Does your clinic allow convenience euthanasias?