"Very little economic activity can be bottled up and saved for years."
That might have been true for hunter-gatherer societies, 10,000 years ago, but since the advent of agriculture, and especially irrigation, human societies have been increasingly dependent on the build up of capital to support an ever more sophisticated civilization. 2,000 years ago, that capital took the form of roads, canals, breakwaters for ports, dykes, walls, structures and, especially, large scale irrigation projects. Modern societies rely on the build up of capital to a much greater extent -- certainly more than I can list in a comment. The capital has been accrued over many decades -- I once read an article in Scientific American that the housing stock of modern Western societies runs on a cycle of about 80 years. The modern build up of capital, on which we depend, would include every form of infrastructure that you could think of, including space-age items such as communication satellites, plus many types that may not be immediately obvious.
Somewhere in the world, each generation must suppress some of its consumption so as to contribute to the accumulation of long-term capital. It used to be that savings were mostly confined to the nation in which they were originated, but of course, over the last 50 years we've seen the globalization of savings, and we've seen vast distortions to the global system, thanks to some countries, such as China, whose government engaged in the systematic suppression of consumption, leading to what Ben Bernanke described as a "global savings glut".
Contrary to what you said, a great deal of economic activity can be bottled up and saved for years. Just because we live in an era in which there was too much saving and too little consumption does not mean that, in the long term, saving is unimportant.
If by "economic activity" you mean activity that supports current consumption, then this is exactly backwards:
"First-world countries are seeing massive demographic shifts towards old age, which means vastly more economic activity must go towards sustaining the retired"
The opposite is true: First-world countries are seeing massive demographic shifts towards old age, therefore the need to raise productivity is more urgent than ever. In the USA, 70 years ago, there were 10 workers for every 1 retired person. In 20 years, there will be 2 workers for every 1 retired person. There we need to see a 500% increase in productivity, over 90 years (measured from 70 years ago), to ensure that everyone can continue to enjoy the same quality of life.
In the past, the accumulation of capital played a large role in raising productivity, and therefore in the future capital might again play a large role in raising productivity. A vast campaign of investment is necessary, to raise productivity. Such a campaign is made easy in the current era thanks to the the low interest rates that were made possible via the global savings glut.