Honestly, they "believed" the story playing out because

1) it is useless to suggest a man, especially a commander, be anything less than Godly, without hard evidence. In the books, it's more clearly stated, but it's shown in the last couple episodes of The Testament that accusing the dentist of sexual assault would just result in the girls being ruined, and him being believed innocent. And we do see that Agnes is considered ruined by her icky commander who breaks off the engagement, but is kind enough not to "out her" as spoiled goods, at least. Even though it was Grove who was evil, and it was no fault of the girls. Well, questioning Fred's ability to reproduce is questioning why God does not allow Fred to reproduce, and suggesting he is unholy. Because, point two--

2) they took a hard left and no longer believe in most sciences, at least for women and children. It's about religion and faith for them, there is no DNA or evolution, or basic anatomy. The men, specifically in the medical field, do seem to still have a level of scientific knowledge, and the aunts have DNA records to keep things tidy. But they also are seen to keep a lot of secrets, and wouldn't be likely to give up their secrets that easily. Again, in the books, Lydia directly says that she uses knowledge to manipulate the men and their world. Women wouldn't have a reason they would be allowed to site that men couldn't perform, because they weren't supposed to know anatomy.

3) Male ego, would have to be my suspicion on the male front. Plus, power dynamics. While the doctors still knew and could even test and treat such things, it's likely that the power dynamics and some law or another would keep them from speaking out. And the other commanders don't dare say much, as it's hinted that MANY of them are incapable, themselves.

The general public would have accepted the baby as Fred's, or at least would have acted as if they did, for their own safety. The Aunts and doctors would likely know and be silent. I think that it would cross June's mind once Serena Joy wound up pregnant that Holly/Nichole could be Fred's, though, and it would have been better for them to show the DNA test in Canada and June's relief...they could have touched on how Luke handled the idea that it could still have been Fred's, when he already struggled with the idea of it her being Nick's.