Scope of natural proofs barrier
Let $f: \{0,1\}^* \rightarrow \{0,1\}$ be a function, and let $C$ be a class of algorithms working on finite slices of $f$. Every circuit lower bound whatsoever is a proof that $f \notin C$, for some $f$ and some $C$. Consider the "combinatorial property of Boolean functions" ${\cal P}_f$, such that 

${\cal P}_f(f) = 1$ and ${\cal P}_f(g) = 0$ for all $g \neq f$. 

A proof that $f \notin C$ is a proof that ${\cal P}_f$ is useful against $C$, in the terminology of Razborov and Rudich. That is, "usefulness" is totally unavoidable -- there is no way to "fall outside its scope." If you have proved a circuit lower bound at all, you have given some useful property.

Note that, if $f \in TIME[2^{O(n)}]$, then ${\cal P}_f$ is also constructive as well, in the terminology of Razborov and Rudich. So for functions $f$ computable within $E$ but not in (say) $P/poly$, constructivity would also apply to at least one property of Boolean functions that is useful against $P/poly$.

So, Razborov and Rudich is more fundamental than you might initially think.