All people are conservative, their first impulse being to preserve the status quo. There are few examples of day to day life that is so terrible it will not be defended against change. Near all change is resisted, viewed with suspicion, and rouses resentment against the effort of will and thought required. This is the case whether or not the change is positive. The greater the change, the more that people dig in their heels. These highly conservative urges are set deep within the core of the human condition, a part of the primate evolutionary heritage of hierarchy and state of mind.
Now consider that we are proposing to up-end everything to do with aging, to change everything in the trajectory of a life through the introduction of rejuvenation therapies. To change the view of parents and grandparents, to change relationships with all older people, to throw out all long term plans for the future and replace them with different ones. The result will be a world made enormously better, in the sense that the disease, suffering, and slow death of later life will diminish rapidly and eventually go away entirely. Yet people are genuinely slow to buy in to this vision: it is a struggle to discard an accepted, known certainty (even if it is of aging, pain, and death) in order to take on the unwanted effort of engaging with future change (even if it is an end to that aging, pain, and death). So people stick with what they know.
This is a deep and serious flaw in our species. Our inherent conservatism strives to kill us in this era of rapid progress in biotechnology, by encouraging us to reject the greatest and most beneficial applications of new life science technologies. It is possible to bring an end to aging in the decades ahead, but that will require the sort of massive funding and widespread support that attends efforts to treat cancer. It requires an acceptance that the new status quo - for now - is to live in a world that strives for healthy longevity, in which the future of a life has an uncertain and unbounded upside. Everything changes for the better, but all planning and assumptions must be reworked. This sweeping change in the public view of aging has yet to happen, and as a consequence funding for rejuvenation research remains anemic.
One of the reasons why the idea of rejuvenating people isn't all that easy to sell is that it challenges the status quo. For good or bad, we're used to the fact that our health goes south on us as time goes by, ultimately killing us if nothing else does. That's not a nice certainty to have, but our species is one of planners; we tend to prefer bad certainties to uncertainty. For example, some people want to be certain that, at some point, they won't be fit for work anymore and will need to retire; they prefer this over the uncertainty of not knowing how they'd make a living at age 150.
That's not the only reason. Radical change requires radical rethinking of anything affected by the change itself; as rejuvenation would affect our social contracts, the job market, future planning, our idea of life milestones, of family, what it means to be old, and many other things, it would take a lot of rethinking - which is something humanity generally does only grudgingly and on its own sweet time.
Think about it: "Granny" is more likely to make you think of a sweet, gray-haired lady with large glasses on her nose baking a cake than of an attractive girl out one late night with friends. Yet, in a world in which comprehensive rejuvenation is common, the granny and grandpa that inhabit our collective imagination would simply not exist; rather, you'd find that grannies and grandpas in their late 80s can't be told apart from people in their 20s; elderly would look just as young as "truly young" people, would be just as healthy, and would be engaged in the activities they prefer rather than having their activities limited by their declining health.
This is only one example out of many more new situations that we, having grown up in a world plagued by aging, would have to get used to; newer generations born in a post-aging world would hardly have any problem with it and would probably end up wondering how anyone could possibly have opposed it in the past. Examples like this are different from concerns such as overpopulation in that they don't represent a potential but tangible issue that might arise as a consequence of rejuvenation; people may have problems with biologically young elderly people simply because they're new and unfamiliar ideas, not because they pose any actual problem.