Not sure I understand the problem. When my daughter was born it also took about a month (at least a few weeks) to get it, and since that was normal the hospital provided a temporary one exactly for that reason. SSN also came in mail around that time frame.
The insurance allowed to register without SSN and provide the SSN later as it was normal to not have it just when the child was just born.
I think there was an option to not get SSN when filling the document for birth certificate, but when searching why I wouldn’t want that I realized it was a bad idea.