I get what you mean but 'easy to validate' in the context of NP means 'can be validated in polynomial time'. It seems to me that you imply that edit distance is harder to validate than problems in NP. Edit distance itself is in NP since P is in NP.
> I mean that there needs to be a fundamental difference between the
> difficulty of finding a solution and validating one. There's none in
> case of edit distance.
To see this, consider the following argument:
Clearly the edit distance between strings of length m, n is bounded by m+n (>= d). So if we found a way to transform the string s1 into s2 using d steps, we can construct a certificate of length O(m+n) (worst case) steps, which can be verified in linear time.
On the other hand (according to this result) you need Omega(m*n) (worst case) steps to find such a certificate.