Literally false.

There is a separate protocol for actual high-risk births (involving the use of Hep B immunoglobulin on top of the vax).

https://www.cdc.gov/hepatitis/hbv/vaccchildren.htm

The idea that hospitals can't or won't screen laboring mothers for high risk is false. Otherwise this special immunoglobulin protocol wouldn't exist.

The Hep B vaccine was studied for years in infants and I'm confident it's reasonably safe. But it's also not of any meaningful benefit to the vast majority of infants who receive it. You can give it any point in elementary school for the same overall benefit (that benefit is: if these kids grow up to become fast-lane gays or junkies they'll be protected).