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Maybe because it's the most commonly used terminology in this context? Trying to attribute some hidden opinion to the author's use of a contextually common synonym (instead of the one you would prefer) seems a bit weird to me.

Because when my dog is old and he is in pain and it won't get any better, I want to gently, let him go, to put him down/to rest. I don't want to 'kill' him. That's what the police would do to someone's innocent dog. Words have emotional connotations.

Which absolutely has nothing to do with "animal pain and animal exploitation", which the GP believes is acceptable, and was using the killing of their dog, somehow as .. I'm not sure how their argument works, because obviously the way we exploit animals is nothing like the way we would gently send a puppy to their final resting place in puppy heaven.

“Euthanize” implies an often tough decision to end a condition of prolonged suffering that has little hope for improvement. “Kill” does not imply any such decision

English is not my first language, sorry. You're completely right.