I would certainly hope that such a reality will never come to pass, but I suppose it would depend on many different factors, economics, AI progression, morals and ethics of the people (I know while a good chunk of the world is becoming more secular). I heard arguments from both sides and I don't think either side is wrong but simply depends on the factors that @Forever Sleep has mentioned.
Assuming that economics is the main driving force and that AI has become more advanced to the degree that it makes many more jobs and positions redundant, obsolete, and people become more and more irrelevant for the economy (barring a few key positions to maintain hardware and software of the AI, jobs that AI cannot completely replace, etc.), then yes, the government could be looking for ways to unburden the system. Ideally, this would be beneficial for people who don't wish to live anymore because if they allowed ease of access to voluntary euthanasia, a good chunk of us would likely meet some/most of the criteria to gain access to it.
However, if the other scenario is true, then indeed, it would be much, much harder to CTB due to the limited access of (reliable) methods and means to effectively CTB. While there are always going to be very determined people who will CTB, the limited means/methods of CTB'ing still available in such a dystopian scenario would be less reliable, leaving to violent deaths and/or drawn out suffering until the end. Hopefully, this scenario never arises and that enough (sensible) people will push back against governmental overreach as well as the unethical prolonging of natural life (ironically, for the religious people who like to claim 'God's plan' and allowing it to run it's course, they would be in contradiction with their own faith if they decide to prolong life against nature-- but that's another point for another discussion).