So far, there are no Chinese voices yet. So, here is my voice.
I learn a lot about my own family history from my father and my father's older sister, as well as my mother and her side of the family.
No dead children. Every character has ~10 siblings, and consequently
~100 first cousins on each side. This option looks a bit crazy.
This is actually not that crazy. My paternal grandmother's mother (my great grandmother) married her cousin. The cousin-cousin relationship resulted in one daughter that was born first, eight sons that all died in childhood, and one daughter that was born last and became my own grandmother. My grandmother and her older sister had a 16-year gap; at the time, it was a generation gap. My grandmother's older sister married and had children, but all her children died in childhood. Meanwhile, my grandmother had four children - two sons and two daughters; and every time these urban-dwelling sons and daughters were to visit their aunt in the countryside, the aunt would proudly introduce them to neighbors. She treated her nephews and nieces as her own children. In the Chinese language, an aunt (姑妈, actually referring to the father's older sister，姨妈, actually referring to the mother's older sister）has the word 妈 in it. In a way, the aunt is like a mother. She's from the older generation, the parental generation. My aunt once told me that she wanted to take care of her own aunt in the aunt's old age, but when my aunt was fully grown, her aunt died, presumably of stroke. All I know is that the woman experienced bad headaches. My paternal grandparents worked as factory workers in the city and had four children. According to my aunt, having only four children was considered unusual at that time. Most families had more than four, all crammed inside a tiny room. But with only four children in the house, there were more resources to go around for each child. If children didn't die of natural death, then they might die of famine.
All this stuff takes place in modern China, because modern China started after the fall of the imperial dynasty. At first, modern China was ruled by the Nationalist party in the Republic of China era, but nowadays, this party is located in Taiwan. Then, modern China switched to the Chinese Communist Party in the People's Republic of China era, and this sovereignty continues to this day. Up to the 1980s, much of the Chinese population lived in rural areas. In rural China, agriculture was based on human and animal labor, though I think this is rapidly changing, as China becomes a fully industrialized country in the future. Having a lot of children was normal, because the children could help run the farm. My paternal grandfather's side of the family would live in a cluster of houses in the countryside. Then, the entire family moved near the big street for convenience. They helped each other build houses from scratch too, and their numbers are in the hundreds.
Characters have dead siblings, somewhere in the time-skips they also
lose children. All of this happens off-screen, and only gets mentioned
in passing. Does this risk alienating my readers?
A Tree Grows In Brooklyn has one character with a lot of dead babies.
Losing a child actually happens on-screen, within the melange of all
the other events (but not taking central stage for very long). I
address the tragedy and getting over it. Again, what would be more
alienating - the previous option, or this one?
I agree with the opinion that you should definitely show how a character deals with losing a child. The emotions can reveal what the character is like.
People just seem to have a smaller number of children, no explanation
given. This option is not too realistic, and always stretches my
suspension of disbelief when I see it, but maybe it's necessary?
This option is actually not that unrealistic. Refer back to my family tree. No one has told me why my grandma only gave birth to four kids in a time when having four kids was considered "few". No explanation is given.
Are there any options I am missing? Or anything about the solutions
presented that I am missing? Which solution is preferable?
You may be missing consanguinity. In the past, Chinese families would want to maintain friendly ties between themselves, so they would arrange the marriage of their children even before the children were born. If the child of one family was a son and the child of another family was a daughter, then they would be paired. If both families gave birth to sons or daughters, then they would not be paired. This led to some consanguinity and inbreeding problems.
You may also be missing out recognizing daughters as part of the family. Historically, daughters were not considered part of the family. They were raised by the natal family for another family. Family records would never mention women, because they would just marry out. A family with only daughters was a family without children. But that changed dramatically in the beginning of modern China, and women were finally liberated. I think this also applies to Jewish families a long time ago, where Jewish families would just not record women in the family records.
My particular story is set in a royal court, and spans several
decades. Under those circumstances, the number and health of royal
children become important both to court politics and to international
Well, if the family is royalty, then the family probably wants to keep power and wealth within the family.