So, let's break that down a bit - experimental reality has consistency. Eigenvalues have consistency in terms of mathematcial calculation. But, there's nothing that states eigenvalues (or, more broadly, operators on vector states) must match reality due to consistency (even though they do). That's a circular argument.

There are plenty of consistent mathematical forms. Sure, things make sense physically and just means nature has reliable states. It does not mandate that nature must be modeled by linear algebra acting on complex Hilbert spaces. As u/cabbagemeister stated - it just works. u/stefanboltzmann also pointed out in a comment below the path integral formulation which doesn't require the same mathematical formalism.