Most trivial neutral pion decay Literature states neutral pion decay by QED cannot occur directly because the pion is uncharged.

However, I cannot see why Photons are not allowed to interact with the Quarks  directly.

Please elaborate which statement is false and why.
 A: A comment which answered the question:

Your quark diagram is incomplete. You skipped the vertex $\pi^0 \bar{u} u$ involving a pion resolving to this virtual pair of quarks, and ending up with the type of triangle diagram similar to  the old-fashioned  one involving virtual Ks or ps. Remarkably, when you include all u and d quarks of all colors, you get roughly the same rate in the hadron triangle calculation, as in the quark triangle calculation.

Do the hadronic processes contribute/exist or are they just of historic meaning?

The hadronic processes are not used anymore, and , as such, they are historical vestiges, superceded by the quark calculations. Nevertheless, as an effective low energy calculational technique, they never had their death certified: lots of alternative techniques can live side-by-side in complementarity. The states in the triangles are virtual, so "exist" is a charged and evanescent fighting word!

A: To describe the uncharged pion in the initial state you will have to project the open $u$ or $d$ quarks onto the mesonic spin zero wavefunction carrying quantum numbers of the $\pi^0$.  At the Feynman diagram level, as mentioned in the comments, you obtain the famous triangle diagrams via  Yukawa interaction describing the decay, driven by the well known chiral anomaly.
The hadronic mediating diagrams can be viewed as a 'course-graining' or effective description of the underlying quark resolved diagrams. The pion decay rate is independent of the description used under the so called anomaly matching condition, but that is a detail.
A: Technically the quarks can meet and annihilate but it is never shown that way because it is harder to do that. For example electron positron antihalation requires a loop like this too. Even if the electron and positron annihilate directly they are going to show it indirectly annihilating to two photons. It is a similar situation. However the Feynman diagram could be drawn were the quarks meet and annihilate into two photons.
