That's a bit different, and as it should be, but then it's likely clearer to just directly state the intent to protect the future generation.

"Family first" in my experience has always referred to the lateral or upstream bloodline. Deadbeat parents or lazy piece of crap siblings trying to justify grift or outright abuse.

I'm an ideal world, the phrase wouldn't exist and people would just be decent; parents would prioritize their children and siblings etc would just help each other. I wish this were the case, but as with many others I've instead come to trust and rely on found family instead of bloodline.

And I'm forever in debt to my daughter. I will never expect her to put my life or needs before hers. My disgust at my own parents' egotistical failings only magnified after personally experiencing becoming a parent.

Family is a sorting category word though, so while there's truth to kids first, the phrase regarding family first is very very often abused to try forcing the hand of broken ties between the aging parent wanting the child to instead provide regardless of whether it's deserved or not. It's frequently an excuse for abusive siblings who've continually taken advantage of their kin.

Bluntly, relation is pure chance and does not immediately imply a debt with exception from "parents must care for their child" by social and biological need. I don't owe my physically and mentally abusive parents or brother a damned thing except my disgust.

People in general, not just kin, need to earn respect and community with each other. The bond of parent caring for child doesn't immediately imply the child owes the parent for "bringing them into this world", and instead implies a larger scope debt toward that child then being indebted to any children they subsequently bring into this life.