The fear of population collapse is about protecting the established economic system which relies on an eternally growing customer/worker base. The economy cannot survive a long-term shrinking population.

The secondary issue of how to care for the elderly when they outnumber the young can be solved by simply not caring for them. (Not my recommended solution but that's what will happen if the situation arises).

Life extension won't solve what governments are worried about. While it lowers the health care costs of looking after the elderly, it does little to increase the birth rate of new customers/workers.