Note that as savings pile up in banks and brokerages, they supply the world with more loans for starting new businesses or new ventures. The money in a bank doesn't just sit there. It is a basis for economic activity throughout the country and even the rest of the world.
A more appropriate example is not an island that constrains population, but a population where new people are born and make investments into new businesses and new ideas. You can certainly have people transition more towards savings over time as they enter their prime career earning potential, without it impacting the rest. This is because there will always be other people who are willing to start businesses.
I addressed that by saying the islanders could work on infrastructure and labour saving tools while young. That's investment, deferred consumption.
However, this has limits. A country with a 8:1 worker:retiree ratio is currently much more productive than a country with a 2:1 or a 1:2 ratio.
Maybe savings could change this for the future, by really making something that hardly needs labour. But presently we have a real labour barrier to this kind of saving.
I agree it would be better if we saved more though, as this does indeed translate to invest. I hadn't explicitly spelled this out, so thanks.
The islanders are indeed better off if they defer. But, they cannot make themselves so well off as when their society was young, and they still need some labour.
A more appropriate example is not an island that constrains population, but a population where new people are born and make investments into new businesses and new ideas. You can certainly have people transition more towards savings over time as they enter their prime career earning potential, without it impacting the rest. This is because there will always be other people who are willing to start businesses.