[Mila]

"It has at least has some embarassing information like “Muslim” or “Father: [someone else]”"

I have never heard of a birth certificate where a religious denomination is required. As for the name of the father, since you are not under oath when filling out the form, you can put any name that you want or you can put none at all.

[MrB]

I’m being charitable with the assumption that the BC would say he was eligible for POTUS,

and trying to come up with an explanation as to why he’s fought so hard to conceal the info in his BC.