> If we were immortal, the birth rate would decline

Would it? In modern, low reproduction societies the dominant interpretation of the universal "honour thy father and thy mother" focuses on care. Even the slightest resemblance to a chain of command that might be present earlier stops at adulthood. But with people forever overshadowed by their undying ancestors, we might see a massive resurgence of the competing interpretation which sees it as a natural power structure. If that happens, well, welcome to pyramid scheme hell.