Why would gravity behave different than 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).
So, to associated something with a symmetry group of SU(1) would be effectively the same as not associating it with a symmetry group.