What does the special form of the base point of secp256k1 allow?
One thing we know is that it doesn't allow any special attack.

Let us assume that we knew a special point $G$ that makes computing the discrete log easy, that is, given $xG$, it's easy (or at least, easier than expected) to recover $x$.

Then, given $H$ and $xH$, here's what we could do; we know that $H = yG$ for some $y$ (as $G$ is a generator), and so we can use our special purpose discrete log problem to recover $y$.  Similarly, we have that $xH = zG$ for some $z$, again, we can recover $z$.  So, we have $x(yG) = zG$, or $x = y{-1}z$; that immediately gives us the value $x$.

Hence, if they knew a 'weak point' $G$, that would imply that the entire curve is weak.  Since we don't believe that, we can assume that their special value $G$ is no weaker than any other point (as long as $G$ generators the entire group, which is true in this case)