Yes, I think it's a problem for society. Death in old age frees up social, economic, physical, and political resources for the next generation of the living. If the rich and powerful escape death, because after all they will the people with the resources to do so, society will lose the adaptability and natural change that comes from new generations taking the reins.
I assume you mean that dying is the most terrifying pat of the natural human experience. Also, I'm not sure why you infer that me thinking death is a natural part of life, means that I'm happy or eager to die.
There are many reasons that people living forever would be a problem for society, the most obvious being an ever-increasing population.
Fertility rates are below replacement, which means that population sizes are convergent. A decreasing population is a more likely future scenario for many western countries, even if human lifespan was indefinite.
Fertility rates are currently below replacement, there's no good reason to imagine they will always be that way, particularly after global population numbers peak and fall to, say, half or a quarter of their peak.
Because living longer is a huge drain on resources that could be better spent on other things. End of life care is expensive and rarely results in a "good" life for the the life being extended.
So I think curing means basically opt in death or something like that. Right now extended life is bad because the person isn't in his prime but curing aging is basically gonna keep him in his prime. This might be what they meant.