Wut? Plenty of countries have universal health care but most don't currently allow euthanasia or voluntary assisted dying, even for terminally ill people in chronic or unbearable pain with only 6 months to live. (It's playing God lol.)
It's a crime in most places to even assist a terminally ill person to end their own life even if they're fully mentally competent and administer the drugs themselves. Can get you up to 14 years in prison in the UK and Ireland and lesser prison sentences plus fines in France and elsewhere.
That debate has been going on for decades (long after those countries had universal health care) and proposals keep getting rejected whenever it gets to parliament, often citing the "slippery slope" argument and the Hippocratic Oath.
Although public opinion in a lot of places has actually shifted to show a majority of the public in certain countries now support it for specific cases.
(We do it for our beloved pets when they're suffering and no one has a problem with that, even though an animal isn't capable of consent tbf. Guess they deserve more humane treatment than humans, who have to suffer in agony til the bitter end and die in pain (or resort to secretive options without being able to tell their loved ones and say goodbye properly) because that's Nature's way, and humans have of course never done anything in history to tamper with nature to alleviate our suffering. Who needs medicine? 🤔)
Only a handful in Europe including Switzerland, Belgium, the NL and Spain, and other examples like Aus/NZ have relaxed the rules. And even then it's pretty recent in most cases, while universal health care has been a thing for years.
TL;DR: Universal health care doesn't automatically lead to bumping off grandma to get her life savings.