Is there a simple argument why graph isomorphism is not NP-complete?
The type of argument you are looking for is as follows:

If graph isomorphism were NP-complete, then some widely believed complexity assumption fails.

There are at least two such arguments:

Schöning showed that if graph isomorphism is NP-complete then the polynomial hierarchy collapses to the second level (equivalently, $\Sigma_2^P = \Pi_2^P$).

Babai's quasipolynomial time algorithm implies that if graph isomorphism is NP-complete then the exponential time hypothesis fails.


Whether such arguments are &quot;valid&quot; or not is a sociological question, not a mathematical one.