Respectfully, you're missing the point. It is often the best (and only, in my mind) choice, and it is always sad. It's not the word or even the concept that's the problem. Using it in a post about euthanasia is fine; using it as callous euphemism for shutting down your business is pretty terrible.
It also might serve as a hint to the emotional connection they formed while building their enterprise.
You could view euthanization as desirable and good. As mournful and difficult, but good nevertheless. And considering humans specifically, of your own choice, under the supervision of a team of supportive physicians (and friends and family).