But without it, you get no household registration as a married couple, which means no birth certificate, which means no household registration and lifelong non-personhood for the child
Where in the world are you getting your ideas? You can't (legally) prevent a child from being registered for a birth certificate even if you want to, as you would expect from the fact that birth certificates are meant to benefit the government, not you. It is literally the first element of Texas HSC §192 ("Birth Records"):
> Sec. 192.001. REGISTRATION REQUIRED. The birth of each child born in this state shall be registered.
Children receive birth certificates regardless of whether their parents are married, or are citizens, or any other consideration. There is no requirement for a marriage certificate, or for anything else.
It's also not true that you can't be registered as a married couple without a marriage certificate. You can, and it's easy.
Edit: You're probably talking about China, not Texas.
