Here is something quick, as I don't have time. I didn't reply because it wasn't clear to me what the original question really is. But here are some quick comments.
Of course the WZW model famously exists and describes strings propagating on a spacetime which is a group manifold. Via the FRS theorem and related facts the WZW model is one of the mathematically best understood string models. As such it has received a great deal of attention.
Of course it is not critical and needs to be combined with more stuff to make a critical string background. I guess that's what the question is wondering about, and the answer is: sure! That's what one considers all the time.
The only trouble is that KK-compactification on group manifolds is not quite realistic.
On the other hand, the chiral half of the WZW model, also known as "current algebra", appears all over the place in realistic models: namely as such the WZW model appears notably in the heterotic string -- and hence all over the place in string theory. See also at parameterized WZW model for a geometric interpretation of the Green-Schwarz anomaly cancellation of the heterotic string in terms of "bundles of WZW models". The discussion there is the most general realistic answer to how WZW models and quantum gravity appear jointly. (I should expand on that, but not right now.)
Another story is if we allow ourselves to pass to WZW models for "higher Lie groups", hence for the global objects which Lie integrate Lie n-algebras . Doing so one finds that all the Green-Schwarz action functionals for superstrings and super-p-branes are WZW models --- on super-spacetime. One can classify all these super-Lie-n algebra WZW models and finds... the full brane scan structure of M-theory:
On the other hand WZW-type superstring field theory is something different. This is not about strings propagating on group manifolds (well it is about strings generally, so it will also include strings on group manifolds) but is instead about a way of formulating "string field theory" (second quantized strings) in a form that exhibits a kind of "second quantized WZW term".
Actually string field theory maybe more naturall involves a "second quantized Chern-Simons term", namely a Chern-Simons term for a Lie n-algebra as \(n\to \infty \). (Everything is higher Chern-Simons, that flows right out of the foundations of mathematics...)