You also might think about how things are changing ignoring the birth rate. People are living longer after retirement. This means that even without a change to the birth rate, there are fewer workers per retiree. Since retirees consume but do not produce, this means that per capita growth will be slower even if per worker production growth stays the same or increases.
Now add in a dropping birth rate. A dropping current birth rate means that in twenty or thirty years (thirty is an extreme case, e.g. a doctor or lawyer), we will have fewer workers per retiree. The dropping birth rate that happened thirty or more years ago is reflected in fewer workers now.
Given the increases in longevity and the comparative lack of increases in the retirement age, the economy needs an increasing birth rate to compensate. An alternative is an increase in immigration.
If we go back and look at the supply and demand curves, your original hypothesis was that increasing the supply of labor would lower prices while increasing demand would increase them. But in this situation, demand is staying high due to retirees while supply is dropping. And of course, a higher birth rate now won't help with that now. It takes twenty to thirty years. A higher birth rate now would increase consumption (children have to eat, etc.) without immediate production benefits. But some people worry about what will happen in the future.