(Apart from that, I'd say that the caricature you're depicting is not particularly good at responding to people in a productive or endearing way, which is unrelated to the problem itself.)
Nick Bostrom's article is advocating exactly that:
Instead of a massive publicly-funded research program to halt aging, we spend almost our entire health budget on health-care and on researching individual diseases.
He seems to be saying that if we halt ageing, we'll stop dying from other disease, or in any case that ageing is more important than any other disease.
(3) Administration became its own purpose. One seventh of the economy went to dragon-administration (which is also the fraction of its GDP that the U.S. spends on healthcare). Damage-limitation became such an exclusive focus that it made people neglect the underlying cause. Instead of a massive publicly-funded research program to halt aging, we spend almost our entire health budget on health-care and on researching individual diseases.
He's equating the spending on health care with the "dragon-administration" that he describes as a pointless, misguided task. So he believes we shouldn't be spending that money on that sort of task, i.e. the US should not be spending a seventh of its economy on healthcare, because that's just "damage limitation".
He's further saying that instead of "a massive publicly-funded research program to halt aging" we're spending that money on "researching individual diseases". In other words, he thinks that that money would be best spent on that "massive publicly-funded research program". Else, what's the meaning of "instead"?
Bostrom's belief is that halting ageing will cure all other disease. According to his allegory, ageing is the one big disease that kills everyone eventually. So if we cure it, we save everyone. Therefore, we should be working to cure ageing and abandon all attempts to cure all other diseases. That's the morale of the story: don't bother with tigers and snakes ("individual diseases"), don't bother with dragon-administration (healthcare), just kill the dragon, save the world.
Note that he's saying all that quite straight-faced, completely ignoring infant deaths (5.6 million under-fives died in 2016) and deaths of people in young age, i.e. many millions of deaths that have nothing to do with ageing and that a magic immortality pill will never get the chance to help in the first place, because they will be dead long before it can stop them from ageing. He's not explaining how a cure for senescence will cure or prevent infections, or genetic diseases, either.
His whole point is completely illogical, irrational, and it's obvious that even people who broadly support it have not really realised what the heck that guy's talking about.
There certainly exist quite a number of diseases that have nothing to do with aging. On the other hand, there also exist quite a few health problems, including many that represent a substantial fraction of healthcare spending, that do relate to age-related degeneration. Taking that into account would produce a better allocation of resources.
Of course, multiple factors can contribute to that, including induced "laziness" in large sections of society due to infiniteness of life.