Any reason you are specifically picking the Higgs field in your question? I am asking, because this sounds a bit like you are riffing on a common misconception that the Higgs field has something to do with gravity, which is not the case. The interaction with the Higgs field is the reason some (only some) of the particles have a mass, but explaining gravity does not need to have anything to do with the Higgs field.
But yeah, it is fair to ask why gravity should behave any differently than any other quantum field -- in the context of Quantum Field Theory (one of the two incredibly successful theories of physics) that is a great question. One handwavy reason it seems different is that in General Relativity (the other incredibly success theory of physics), gravity has to do with the geometry of space and time, not with what other fields exist in that space and time (and as such is explicitly different than the other fields).
My understanding is that the Higgs field should be simpler than gravity because it's just a static value whereas gravity is SU(1)? At least to me it seems logical that if the Higgs is quantized, surely more complicated fields would be as well?
SU(1) is the trivial group. SU(n) is the group of unitary nxn matrices with determinant 1. There is only one 1x1 unitary matrix with determinant 1, and in fact there is only one 1x1 matrix with determinant 1 period, namely, the 1x1 matrix whose only entry is the number 1.
So, to associated something with a symmetry group of SU(1) would be effectively the same as not associating it with a symmetry group.
But yeah, it is fair to ask why gravity should behave any differently than any other quantum field -- in the context of Quantum Field Theory (one of the two incredibly successful theories of physics) that is a great question. One handwavy reason it seems different is that in General Relativity (the other incredibly success theory of physics), gravity has to do with the geometry of space and time, not with what other fields exist in that space and time (and as such is explicitly different than the other fields).