As much as we like to pretend to be above such things our psychology proves that time and time again at our base programming we are pack animals. That does not mean we are slaves to these impulses. We are still creatures of reason. It just means it is just as unwise to ignore that these impulses exist as it is to constantly give into them. That means that we tend to organize ourselves into hierarchical groups whenever we are around each other be it adding a new friend to a circle of buddies or a new coworker at the office. Families tend to be a little more subtle about this power dynamic but mostly because such jockeying within the food chain of a household tends to be either instantly determined by birth or build up over the years. The only real exception to that rule is when a new family member is added to the mix. This is one of the few times a person can earn or lose their place in the household so quickly.
As the newly adopted daughter of the von Einzbern family Chloe has one chance to determine who is boss. She clearly can’t overtake Irisviel. Humans can not pass into the realm of the gods. Kiritsugu is just too much of a stone cold bad ass for the discount Archer to overtake so him so the only point of entry on the top is to try to sneak in above Illya.
Who will be on top when the dust settles? Will Shiro always be the bottom of the totem pole? (Spoilers: yes.)