I was wondering why exactly QG failed at being described by standard field theories and even QFT. Where exactly do the infinities come from/arise...

or is it that each QFT is unique to the force which it describes..

The standard methods of perturbation theory, in particular perturbative renormalisation fail as Newton's constant has negative mass dimension. This makes it impossible to use in standard perturbative procedures. In essence this will mean that an infinite number of parameters to control the perturbation series will be required. This, as we understand QFT means that the theory is not well defined by a perturbation series.

 

You can now either consider quantum general relativity as an effective theory. That is one introduces a cut-off and considers things up to one or two loops. In essence you hide away our misunderstandings of gravity at high energy. You can get S-matrices and scattering amplitudes in this way.

 

The other possibility is that quantum general relativity, or something very similar has a well defined non-perturbative formulation. This requires a UV-fixed point under the renormalisation group flow. Steven Weinberg has proposed that gravity maybe asymptotically safe, which means that it is well defined but not as a perturbation series. As particles are a perturbative notion, quantum gravity may not be a theory of gravitons at all!

 

The other possibility is that one has to use other symmetries to render gravity well defined. For example string theory seems to give a perturbative definition of quantum gravity.