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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?

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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.

the way the quote reads it sounds like this physicist is saying that gravity is not a field at all.

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