As far as I know, in every state in the US, the woman’s husband is presumed to be the father unless he can be shown to be outside the country for six months on both sides of the conception. The time period may vary by state, but the legal principle is the same, and it does not change if the father predeceases the birth.
It’s only a problem if 1) the couple were not married, or 2) the baby was conceived by AI after the father was deceased.
I’ve known people in the US in both situations.
In the first, there was a complicated bit of maneuvering involving a voluntary DNA sample from the father’s brother. The father was the only brother, the DNA was not a close enough match for the brother to be the father, but both the Y-chromosome and mitochondrial DNA matched, so it could not be anyone else. If there had been more than on deceased full brother, I don’t know that they could have established it without an exhumation.
In the second, the parents had been married, and the birth certificate was not an issue-- it was the child’s right to social security survivor’s benefits. She won.
I’m not sure that the people in the hospital where the child was born knew the child was AI, and that the father predeceased conception. They knew only that the parents had been married, and no one was making any issue of anything. No one asked to see a death certificate, nor asked if the couple were ever divorced. And once a man’s name (especially a husband’s) is on the birth certificate, it is hard to get it off.
In the case of women who claim the father was Mick Jagger, or George W. Bush (FWIW, I know a guy named Andrew Johnson, but not THAT Andrew Johnson), they may get that name on the birth certificate, but they don’t get child support from rock stars or former presidents without proving it was THAT Mick Jagger, or THAT George W. Bush, and the burden is probably on the women to prove that those men were even in the same state as the mother during the relevant time.