You can't look at a single generation only with humans. The evidence for this is menopause and the fact that humans live as long as we do.

Under a single generational model, there is no evolutionary reason for us to live so long past fertility. Populations where the older generation stopped competing for resources by dying would outcompete populations with the additional load.

Thus with humans it makes sense to count descendents at the time of death.

I believe the reason is because reproduction is replication of information. For most life that information is nearly entirely genetic. Or better put, molecular. Get enough molecules into a baby horse and you have an adult horse. One that can do all the horsey things.

Humans aren't like that at all. Get enough molecules into a baby and you get an idiot.

Humans also need to replicate all the social knowledge that allowed the parents to navigate highly complex rulesets inherent in a functional society. This takes at least 20 years of education and nurture. I believe Grandparents play a critical role in passing on the ideas and traditions that enable grandkids to make the tough choice to have children of their own that will want to have children as well. Etc. etc.

Blame the invention of language for starting the evolution of a new emergent layer of life called society, humans are different in kind.