I’ll add this: If I could make the decision for others to enter the Kingdom of Heaven, I would do so for ALL. But no one can make that decision for another. I cannot repent and believe on behalf of my wife, nor can I do so for an infant anywhere.
I can, as head of the household, decide my house will be run on the principles of God. In that sense, I CAN save my entire household - in the sense of sanctification (separation from the world):
“If any woman has a husband who is an unbeliever, and he consents to live with her, she should not divorce him. For the unbelieving husband is made holy because of his wife, and the unbelieving wife is made holy because of her husband. Otherwise your children would be unclean, but as it is, they are holy.”
But that talks of sanctification, not justification - and it may well be the context that talks about an entire household being “saved”. Justification is required for someone to become a citizen of Heaven, to board the SS Jesus and take a cruise whose destination is to be conformed to the likeness of Christ. But you can be made different (separate / holy) by the decision of another to have his home abide by different principles.
As noted in Acts 16:
“And he rejoiced along with his entire household that he had believed in God.”
His household rejoiced that he had believed...
Read that again. Slowly and let it sink in.
And it goes deeper than this (covered this in post #145).
The promise the disciples made in v. 31 was that the entire household would be saved. What's very interesting is that the only recorded appeal to believe is to the jailer himself -- NOT the household (v. 31).
And to confirm this, the ONLY recorded response of belief is the jailer's belief response -- v. 34. NOTHING is mentioned about the rest of his household believing.
Yet the entire household was baptized (v. 33). The entire household rejoiced (v. 34).