You say:
"This issue is addressed directly by Ionel-Parker in their paper on relative GW-invarianst and also by Li-Ruan in their virtual neighborhood construction. Both come to the conclusion, that after a generic choice of almost complex structure, such behavior can be ruled out."
but I don't think that you have correctly characterized their solution to the problem. You do not get to dismiss the behavior --- instead you deal with it in a way that maintains transversality by caveat. In order to explain, it is helpful to compare the way the problem is handled in algebraic geometry and then compare it with what is done in symplectic geometry.
In algebraic geometry, if you want to have a proper moduli space of relative stable maps to $(X,D)$ and maintain the condition that components of the map are not mapped into the divisor $D$, you are forced to allow the target $(X,D)$ to degenerate a singular target $(X,D)\cup\_D F$ consisting of the union of $X$ and $F=\mathbb{P}(\mathcal{O}\oplus N\_{D/X})$ glued along the zero section of $F$ and more generally, you need to allow a finite chain of copies of $F$ glued successively along the $0$ and $\infty$ sections: $X\cup_D F\cup_D\cdots\cup_D F$.
Although it is perhaps a little hidden, this same phenomenon is forced on you in the analytic setting of Li-Ruan and Parker-Ionel. Instead of the singular target considered above, they use a target of $X/D$ with a metric that looks like an infinite very, very long tube in the neighborhood of $D$. When they analyze the limiting behavior of maps to such a space, they see a similar phenomenon cropping up: the energy of a map to the tube partitions itself into sections so that in the limit, you can view it as given by a collection of maps to a bunch of long tubes, each of which give you a map to $F$ (the point being that the infinite tube is conformally equivalent to $F$ minus the 0 and $\infty$ sections).
There is a picture of this on the top of page 5 of this paper of Parker-Ionel:
http://arxiv.org/pdf/math/0010217
They are actually illustrating the gluing of relative invariants (so that manifolds appear on both sides of the tube instead of just one side), but you can see the appearance of multiple copies of $F$.
So to answer the question: there is no oversight in those Parker-Ionel, Li-Ruan papers. The phenomenon that McDuff is illustrating is a fact of life for relative invariants, even if it is manifested a little differently in the analytic setting.
I am not familiar with symplectic geometry so let's assume everything here is at least K\'ahler.
If $g=1$, then the condition $\langle [pt], [pt], \ldots \rangle^X_{1, [C]}\neq 0$ implies that the variety is uniruled, which is equivalent to $\langle [pt], \ldots \rangle^X_{0, {C}}$.
I hope it is true that for a rationally connected fibration over a curve of any genus, your condition $\langle \beta, [pt], [pt], \ldots \rangle^X_{g, [C]}\neq 0$ is always true. And it is true when the fiber dimension is at most $2$. Basically as long as you know that there is a section which gives non-zero GW invariant, you can glue this section with curves in a general fiber which is minimal among all curves with non-vanishing GW invariant $\langle [pt], [pt], \ldots \rangle$.
For ruled surface, what you said is true. The methods used in the paper here certainly work.
Best Answer
The examples in Ruan's paper "Symplectic topology on algebraic 3-folds" (JDG 1994) seem to qualify: take any two algebraic surfaces V and W which are homeomorphic but such that V is minimal and W isn't. These are nondiffeomorphic, but VxS2 and WxS2 are diffeomorphic, and Ruan gives lots of examples (starting with V equal to the Barlow surface and W equal to the 8-point blowup of CP2) where the diffeomorphism can be arranged to intertwine the first Chern classes, whence by a theorem of Wall the almost complex structures are isotopic. However, the distinction between the GW invariants between V and W (which holds because V is minimal and W isn't) survives to VxS2 and WxS2, so VxS2 and WxS2 aren't symplectic deformation equivalent.