What I really do not understand that Is it the peculiar nature of graviton is creating the problem or the whole model lacks a fundamental drawback in unifying the all four forces of physics?
both because it's the same thing. The Standard Model - and any quantum field theory - may only contain fundamental fields with spins at most 1. Spin equal to 1 is the right spin of photons, W-bosons, Z-bosons, and gluons - more generally, gauge bosons. But gravitons have and must have the spin equal to 2.
Incorporating the gravitational force is the same thing as incorporating interactions involving the gravitational field - the metric tensor - or, equivalently, incorporating the particles that are quanta of this field, the gravitons. This is not possible in quantum field theory. Of course, it's not possible in the Standard Model either because the Standard Model is a quantum field theory and moreover, it is one defined by a particular spectrum of matter and force particles which don't include (and can't include) gravity.
Gravity may still be added but only as an effective field. While the Standard Model may be made finite to arbitrarily high energies, the addition of gravity reduces the "quality" of the model qualitatively because it may only be reliably used for de facto classical calculations.
In practice, this is not a problem because in particle physics experiments, gravity between the particles is negligible. In principle, it is a huge problem and string theory is the only known - and, arguably, the only mathematically possible way - how to reconcile gravity with quantum field theory.