Caring for the needs of your family is a commandment from God...therefore by taking care of your family, that is a part of putting God first. But if a question of loyalty to God vs. loyalty to family should come into play...loyalty to God comes first.
So God's commandment is, 'care for the needs of your family until I tell you otherwise.' Is that fair? I'm not trying to put words in your mouth, so if I'm wrong, sorry. I'm just not sure I understand.
Caring for ones family would include not just physically, but spiritually as well. So it's not really "until I say otherwise". Worship of God is supposed to be something that involves the entire family, with the father taking the lead for the household. (not a Sunday school teacher) It is a father's responsibility to set a good example, care for his family's needs emotionally, mentally, physically and spiritually. And take the lead in teaching the children about God's Word.
"And these words that I am commanding you today must prove to be on your heart; and you must inculcate them in your son and speak of them when you sit in your house and when you walk on the road and when you lie down and when you get up." (Duet 6:6,7)
So even taking care of your family is not to the EXCLUSION of putting God first. Worship of God is central to the family's daily lives. If a family member chooses to not be a part of that any more. That's where Jesus' counsel that you must put God first. And Jesus did point out that putting God first *absolutely would* divide families in that type of a situation. Ideally, it would be nice if the family was united in worship...but everyone has free will. And even Satan was a righteous angel before he rebelled. So even a perfect parent like God cannot stop free moral agents from making bad decisions if that is what they desire to do.