Is it unethical that my moms considering putting her dog to sleep instead of driving 16 hours to the vet?

Her dog is 13 1/2. He has arthritis, and can't walk very well. He gets injections every two weeks for it. He's also very skinny so the vet thinks might have cancer because he still eats a lot. They haven't confirmed cancer though. So anyways, yesterday the left side of his face swelled really bad. No vet near us would take him, so we had to drive 6 hours round trip to an emergency vet, and they determined he has a tooth abcess, or however you spell it. So they gave him some medication of a 10 day supply that will take the pain away, but they said to see if our primary vet here will extract the tooth. If if not then the only vet that will do it is 8 hours away, so 16 hours round trip. My mom said that if our local vet can't do it, then she might have him put to sleep. Is that wrong? I think her line if thinking is that he's old and likely won't live much longer but I think we might be able to get a few more years.

Is it unethical that my moms considering putting her dog to sleep instead of driving 16 hours to the vet?
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