Very little research currently goes into attacking aging directly - as opposed to handling things that are in no small part downstream from aging, such as heart disease. A big reason for poor "longevity gains" is lack of trying.
Age-related illnesses shouldn't be dismissed with "they're just old" of course but there's no reason to expect a single cause. Other than passage of time itself.
That's not a reason to say that cancer is somehow "not a disease". It obviously is. We don't want cancer. We fund efforts to research cancer and funnel money into better cancer treatments, and we get results.
Aging should get the same treatment.
I am not against trying to "solve aging", but I don't think we should think of it as just a disease, and there should be more plans on how to deal with the sudden "infinite" number of humans. While I may want to live forever, I would definitely not enjoy that in all circumstances.
That's beyond optimistic. What's more likely to happen is, we'll uncover some major pathways for aging and find a way to target them to slow aging down somewhat, at first.
People who get anti-aging treatments would live for longer, and would be healthier while they do. The adoption would be gradual, and it'll take a while for them to come down in price and proliferate worldwide - and it would still be up to people to decide whether they want them, although most doctors would recommend they do. The first generations of anti-aging treatments would allow people to live to the age of 100 fairly reliably, and remain healthier and more active while they do. Future generations would improve on that.
There will be no "sudden infinite number of humans" to deal with. Even if we started out tomorrow (for example, if it was confirmed that Ozempic has broad anti-aging effects), it'll take decades for this effect to become noticeable. Humanity can adapt to something like that easily.