Interesting theory that is contrary to any actual observable evidence. Look at any country with an aging and shrinking population.

Correlation is not causation. Countries with aging populations are ones that have undergone the demographic transition. They have low growth because they are already highly developed and the opportunity for increasing productivity is low, not because they have too many old people.

So you are admitting that if a country is highly developed and have undergone a demographic transition they have low growth. You just proved my point, thank you.

No. Your point was that low population growth produces a drag on productivity growth because of an aging population. This is wrong; to reiterate, advanced economies have low population growth because they have undergone a demographic transition - i.e., they have gone from high birth rates and high death rates to low birth rates and low death rates. This is a result of a number of factors, mostly relating to the increasing rights of women. The effect of this is a higher fraction of elderly in the population. However, growth (per capita) is exponential; the fraction of elderly is fixed. Therefore the fraction of elderly cannot be a drag on growth.

The demographic transition is the result of development, which usually brings with it increasing mobility for women and depresses birth rates. Another, independent, effect of development is low growth rates, because a developed economy has less room for productivity growth (i.e., when you give an uneducated person an education and some technology, you can increase their productivity - it's hard to do the same with an educated, tech-savvy person).

To sum up: development causes demographic transition causes aging. Development also causes low growth rates. Aging does not cause low growth rates.