Clarification on eigenfunction property from Oppenheim's Discrete-time signal processing, 3rd ed
This statement can only be understood given its context. Right before the sentence you quoted, we have this statement:

It is no surprise that the condition for existence of the frequency
response is the same as the condition for dominance of the steady-state solution.
Indeed, a complex exponential that exists for all $n$ can be thought of as one that is applied at $n = -\infty$.

Consequently, starting out at $n=-\infty$, if the system is stable, all transients must have died out at &quot;finite $n$&quot; because at finite $n$ the moment that the exponential was applied is infinitely long ago.
We also depend on stability because otherwise the eigenvalue $H(e^{j\omega})$ wouldn't exist, at least not in the classical sense.