Word for giving preference to the oldest child
Seniority is a general concept that applies to people who are (a) older or who (b) have been around for longer. Lexico explains:
The fact or state of being older or higher in rank or status than someone else.
A privileged position earned by reason of longer service or higher rank.
The concept is commonly applied to companies and other hierarchies, including family structures. Within a family, the eldest child may have more seniority than his or her siblings by dint of longer service or age. So in your scenario, when one person says,
"Why do they get to sit in the front seat" "Because they're the oldest"
That primes a response on the basis of seniority, which would benefit the oldest.
"That argument doesn't work. We don't practice seniority."
If you wanted to be more precise, you could even specify the kind of seniority, e.g. "sibling seniority." Here is blogger Jules Kendall's description of seating priority with her siblings while growing up; she describes the power of choice she has over her siblings as "sibling seniority":
Later, the only child sat in the front seat in between the parents if the seat stretched out like a bench. If there was a console, they laid (or sat or twirled) in back seats like royalty on wide expanses of bonded leather or short pile upholstery. One brother, then two. My space became crowded, but as the oldest I was able to skirt the middle seat thanks to irrefutable laws of sibling seniority.
Unfortunately, the word you are seeking appears to be obsolete. It is the use of the word 'firstborn' to express the rights of the firstborn.
Firstborn :
- That is the right of the firstborn. Obsolete. nonce-use.
1770 O. Goldsmith Deserted Village 256 Spontaneous joys..The soul adopts, and owns their first-born sway.
Oxford English Dictionary
I think you will need two words to fulfil your sentence :
That argument doesn't work. We don't practice firstborn preference.