There was a lot of debate about infant baptism in The Church — it raised a lot of questions about what happened to infants who died before being baptised (before modern medicine, many kids didn’t last to see their 5th birthdays). The early Church answered that pretty well..
Baptists answer it better.
#1, there is NO BIBLICAL EVIDENCE for infant Baptism - period. Zero. Zilch. Nada. Saying “and his house” is going beyond what Scripture actually says. Inadmissible because that isn’t evidence, that is eisegesis.
#2, Baptism is not a prerequisite for Salvation. It is a picture of what Christ did for you and a testimony of your faith in Him to the world AFTER you are saved. David’s baby immediately went to heaven which is where all babies go. No need for theological nuancing. Just go with what Scripture says and you are fine.