Interesting that the stigma is so recent, and it appears it is largely self-inflicted: I got the impression that it is mainly the women themselves who choose to hold their "failure" quiet, and not some outside moral authority (though of course societal expectations silently drive the reclusiveness response).
My hunch is that because most women had more children (on average) than contemporary women in developed countries, they either had the miscarriage after they'd already given birth to other children, or at an early age, with still plenty of time to have more children, absent serious health issues.
We probably don't call miscarries spontaneous abortions because the implication is that you just decided on a whim to have an abortion as opposed to what a miscarriage usually means which is an abortion of an already dead fetus.
Also i don't get why death certificates for non-born children is something that people are so angry about. How dare we cherish life and mourn when a death occurs. It seems like the people who are truly offended by such a proposal have some obscure views on meaning of life and death.
In one case the pregnant body decides to kill the fetus and eject it, in the other an outside actor induces the same process. This is spontaneous in the "happens on its own" sense, not in the "likes long walks in nature" sense: It is a technical term.
As for the death certificate issue, they are transparent ploys for political points made without granting any agency to the affected mothers.
I wonder if some people might fear that death certificates would be the thin end of another wedge towards banning abortion. It would also turn something that some people might like to keep deeply personal into a state-registered event; how long until they become compulsory? There are places in the world where miscarriages are seen as suspicous. Potential crimes, worthy of investigation. I think we all know there are groups who'd like to see that in the US.
Fore more thinking on a very related subject, I warmly recommend this talk about the metaphysics of pregancy: https://www.youtube.com/watch?v=COZEansw_Ts