How do we know what would happen? As far as I am aware, euthanasia in the form of assisted suicide is not legal except in a few US states and in Switzerland.
But before we answer that question, for the time being, let's put aside what is going on in jurisdictions where such practices are legal.
I look at what is actually happening because of the current legal situation, and as is evidenced by Scott Adams' experience, there is clear proof of harm by prohibiting assisted suicide. There are many other people out there who share similar painful experiences, whose loved one died in protracted suffering, agony, and pain; who did express wishes to not be forced to live in such circumstances, but for physical reasons, could not terminate their own lives, and for legal reasons, were not allowed to delegate that responsibility to others.
I would consider the addressing of a real and surprisingly common injury to have more merit than a hypothetical or perceived injury. The fact is, these terminally ill individuals are going to die; it is simply a matter of how and when that death should occur, and the individual right to self-determination of that fact.
Now, let's look at what is actually happening with assisted suicide in those jurisdictions for which it is legal. Are bedridden Swiss people suddenly pulling the plug in droves? Is their society collapsing under the collective weight of some Alpine-induced ennui? Are there death panels of doctors killing patients in Washington? Were you even aware that these places allow assisted suicide because there have been reports of the unethical application of this practice? Were there news reports that someone was mistakenly killed off even though they would have lived, and the doctor used the legality of assisted suicide as a legal defense? Because unlike what you are imagining in your head, those fears have not come to fruition. You never actually state what those "things" and consequences are.
And as I have already pointed out, even if those imagined consequences were real, they would need to be weighed against the ACTUAL consequences of the status quo.