@DenisNardin Just a random opinion: I feel like one of the moments S/U clicked for me was when I realized the version for Right/Left fibrations is exactly the statements that $PSh(C)$ can be written as a localization of the category of "diagrams in $C$" by "formal congruences" of diagrams meaning final/cofinal functors. I.e. $Cat_{\infty /C}[final^{-1}] = PSh(C)$.

It has the advantage of being motivated for a "model independent" point of view since it's a way to give presheaves a natural mapping out universal property.

The generalization to Cartesian/Cocartesian is not completely obvious but some of the intuition at least can be carried through...

I actually would be happy to know if there's a neat version of this statement for Cartesian/Cocartesian fibrations. I guess one could just take marked categories and the invariant notion corresponding to marked anodynes but maybe there's something better to say?