It is human nature to be capable of committing acts of great evil or economic self-destruction for years on end, and especially in groups. We are not at all far removed at the moment from large-scale genocides, collapsed kleptocracies, meaningless prohibitions, and more. So it's probably unsafe to assume that no state will outright ban the extension of healthy life via medicine in the future: there are more than enough examples of human collectives acting against the long-term self-interest of all their members for decades, and that becomes ever more likely if those at the top invent the means to profit personally from a widespread destruction of life and wealth.
For all that, I do think it's an unlikely outcome. The more plausible outcome is the one that is taking place right now: great economic harm to the pace and breadth of medical development through heavy, centralized regulation. Enormous, entirely unnecessary costs and very high barriers to entry are imposed on clinical applications of medicine, which ensures that a great deal of possible, plausible research and development never happens. Worse, in a system in which all that is not expressly permitted is forbidden - which is exactly the case for the FDA and similar regulatory bodies elsewhere in the world - radically different new technologies such as the means to treat aging are restricted by default, without any politician or bureaucrat having to raise a finger. The entire system of regulation must be changed to even allow them to be considered: which means more cost, more delay, and more work suppressed because it isn't cost-effective to undertake.
Here the possibility of future restrictions on rejuvenation therapies is considered by someone who is more supportive of the existence of a large state than I am. They see the solution as working within the system, being a petitioner to power to beg for the chance to be free enough to make the world a better place. I'm not sure that this has ever had a good record of success over the long term, certainly not when compared to revolution or the establishment of new colonies far enough distant from the state to be largely free from its bureaucracy:
Most laypeople with an opinion [on] biogerontology assume "[life extension] treatments will be centuries in the future", actual specialists with a medical background tend to be more 'optimistic' and postulate some for of accessibility of these treatments somewhere later this century. I'll abbreviate "Life Extension" as LE and Rejuvenation as RE.
The human that has singlehandedly saved most lives world wide may very well have been Maurice Hilleman. In the late 19th to mid 20th century there was a small number of cynics who insisted that vaccinations (and other treatments intended to make people live longer lives) would contribute to Malthusian overpopulation. It is interesting to realize that many of these objections were based on class-prejudice and racism. People who objected to child vaccinations tended to not like poor people very much, and didn''t want 'their' world overrun by the kind of people they took offense to. These sentiments are by no means dead. A very common objection to the mere realization of RE/LE treatments is that "the world would quickly overpopulate". When quizzed strikingly many people today insist that RE/LE might "have to be declared illegal to avoid an overpopulation disaster". These people seem to be unable to infer comparisons from earlier Life Extension treatments (clean drinking water, sanitation, healthy diets, environmental protection laws, vaccinations) from which they benefited, and regard Biogerontological Life Extension as something different altogether.
The process of development of actual "biological immortality" is likely to be a long trajectory of dead ends and catastrophes. The beta stage of life extension may come with painful episodes and failures. Early adopters may end up forking out large sums of private capital for treatments that may or may not work. If earliest stage regenerative treatments were to emerge in the 2020s it may be decades before these treatments would end up safe, affordable, comfortable and easy to use. What is worse - such treatments don't have a convenient fit in the current medical corporate sector. What does a LE or RE treatment actually do? Does it make people less dependent on other medical treatments? If that is the case many established medical conglomerates may very well vehemently object against these treatments, and declare them "snake oil" or "pseudoscience". It is thus quite likely that on the earliest years of emerging LE/RE many consumers may reject these treatments basing their choices on vicious and deceptive media campaigns.