Have more children to even out the population pyramid?

What? Take the example of a country with 3 old people, 5 adults and 2 children. The government has to pay pensions for 3 and expensive healthcare for 2 (vaccines, pediatrics, etc).

If the adults have more children, we'll have the same 3 old people, the same 5 adults and 3 children. This means one more pediatrist, thus increasing costs.
If the population stays as it is, nothing will change.

Why do the governments of countries with aging populations encourage childbearing? They're not magically displacing the elderly, they're simply introducing a new expenditure. It makes sense if you don't want your country to die, but Japan and China are examples of this and they aren't dying any soon.

It makes sense if you want to keep your country from dying, but this reason is never cited and instead they cite the need to "even out the population pyramid because old people are too expensive to maintain". This doesn't make any sense.

What vaccines are expensive? Pediatrics are also a drop in the bucket compared to geriatric care (chemo, heart surgery, end of life care, hip replacements, morphine, paying someone to change their diaper and make sure they don't wonder into traffic...).

Children will eventually be useful. Old people rarely are a good source of labor, and won't ever be useful again. You can force children to do chores, but grandpa is going to sit on the couch all day and hog the TV.