No, the actual risk is that there are larger bureaucratic barriers to treatment than to the assisted dying programme, so it's easier for somebody suffering to die than to get treatment, when effective treatment is available and (in the absence of those daunting bureaucratic barriers) preferable.
Assisted dying is a horrifying, but probably overall good idea; however, we need to reform our medical bureaucracies before it's something that they can ethically provide.
And on that note, it always amazes me we have plenty of people dying of depression or anorexia, conditions they have for many years and their treatments don't work (in these cases) and yet they are not offered psychedelic therapy because it's still forbidden in many countries (it's changing, slowly). I'm not saying it's a panacea - far from it - just if someone is not responding to other treatments or the treatments is making things worse (like the risk of suicide goes up for some antidepressant medications) then why not try a different option. As you say, bureaucratic barriers.