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.