As a rule, we don't perform (or try VERY hard not to perform) "convenience" or "economic" euthanasia here. In certain cases of very serious behavior problems (intractable aggression, for example), we have euthanized otherwise healthy patients, but those are the only examples. If a client cannot afford treatment for their sick pet, we often look for alternate sources of funding, like non-profit organizations or our own in-house Good Samaritan fund. We have also accepted surrendered pets, treated them, and found homes for them after they recovered.
As the article says, though, it's a really difficult situation a lot of the time. We're fortunate to be a large enough hospital that "adopting" a pet every now and then is a cost we can afford, but not every practice can make that arrangement work, and no vet wants to put a healthy pet to sleep.