That might not be true, if you look at the paper:

"The super stem cells prevent age-related bone loss while rejuvenating over 50% of the 61 tissues analyzed." (including the brain).

What do people die of when they die of 'old age'? There's the 3 pillars: cancer, cardiovascular, neurodegenerative. These are often (but not always) metabolic diseases; i.e. cardiovascular death often arises from kidney insufficiency. If you can regenerate the liver, kidney, etc. indefinitely, a large vector of metabolic disease is probably diminished or disappears.

In the paper, monkeys restored brain volume. They reduced the levels of senescent cells to youthful levels. They increased bone mass. This reduces or eliminates many of the threats that inflict casualties among the centenarian population.

Sure, something else could come up that the monkeys start dying from instead. But, given the way humans and monkeys die of old age—by reducing or eliminating all known threats—it's hard to see how this wouldn't extend lifespan.

This paper doesn't prove that it extends lifespan. So to speculate on that extraordinary claim without extraordinary evidence to back it up is useless. It would be far easier to prove this out on a species with a much smaller lifespan like mice, not to mention cheaper, but so far we're unable to make a mouse live longer than 5 years.

Yes, I'm certainly speculating. It certainly seems that this could be a path to extending lifespan. I think the claim is less than "extraordinary" though. Many teams are working to figure out how to extend lifespan in many species—it seems likely that there will be meaningful progress in the coming years or decades.