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I don't think the solution is a hard alteration to the type chart. I don't want every type to have exactly 3 weaknesses and 3 resistances, because it just makes the game more samey and boring. Having the Ice type be fragile gives it its flavor. Not only that, but something like "Ice should be strong against X" only makes it a better coverage option on non-Ice mons, it doesn't solve the issue with mons which carry the type.
I think it makes more sense to salvage the type by other means. For example, I think Ice-type mons should get some sort of boost in Hail like Rock-types do in sand. Ideally, it should do something to address the 'slow-defensive-ice-type' problem. For example, have Hail induce something similar to Delta Stream, where attacks which would hit Ice-types super-effectively instead deal neutral damage during its effect. Alternatively, it could just boost offensive stats. Hail has always been the worst weather, in part because you need an Ice-type team to take advantage of it, and since it's one of the few things which distinguishes Ice-type mons from other types, I think it warrants a major buff.
Also, Freeze-Dry should be buffed to 80 BP so that it can be a legitimate side-grade to Ice Beam, and Ice types should get more powerful moves like Aurora Veil which are only distributed within the type. It should be more similar to Grass-types in that regard, with a large number of useful moves which have good distribution within the type but almost no distribution outside of it.
